Asymptotic Representation Theory of the Symmetric Group and its Application in Analysis

Translations of

MATHEMATICAL Volume 2 19

Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis S. V. Kerov

American Mathematical Society

Translations of

MATHEMATICAL MONOGRAPHS Volume 2 19

Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis S . V. Kerov Translated by N. V. Tsilevich

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE AMS Subcommittee Robert D. MacPherson Grigorii A. Margulis James D. Stasheff ASL Subcommittee Steffen Lempp (Chair) IMS Subcommittee Mark I. Freidlin ( C h a i r )

(Chair)

ACMMIITOTMYECKAST TEOPMST IIPEACTABJIEHMSI CMMMETPHYECKOM rPYnnb1 EE IIPMMEHEHMST B AHAJIM3E Translated from t h e Russian manuscript by N. V. Tsilevich 2000 Mathematics Subject Classification. Primary 20C30, 20P05, 22D10.

For additional information and updates o n this book, visit

www.ams.org/bookpages/mmono-219

Library of Congress Cataloging-in-Publication D a t a Kerov, S. V. (Sergei Vasil'evich), 1946-2000. [Asimptoticheskaia teoriia predstavleniia simmetricheskoi gruppy i ee primeneniia v analize. English] Asymptotic representation theory of the symmetric group and its applications in analysis / S. V. Kerov ; translated by N. Tsilevich. p. cm. - (Translations of mathematical monographs, ISSN 0065-9282 ; v. 219) Includes bibliographical references. ISBN 0-8218-3440-1 (acid-free paper) 1. Symmetry groups-Asymptotic theory. 2. Representations of groups. I. Title. 11. Series.

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Contents Foreword Publications of Sergei Kerov Chapter 0. Introduction $0. Plan of the book $1. General theory of locally semisimple algebras $2. Characters of 6,, and the Young graph $3. The Plancherel measure of 6, $4. Continuous Young diagrams in problems of analysis Chapter 1. Boundaries and Dimension Groups of Certain Graphs $1. Ergodic method $2. Combinatorial examples of branching graphs $3. The boundary and dimension group of the Kingman graph $4. Stirling triangles Chapter 2. The Boundary of the Young Graph and Macdonald Polynomials $1. Ideals of the group algebra (C[6,] $2. Characters of the infinite symmetric group $3. Generalized Macdonald polynomials and orthogonal polynomials $4. Orthogonalization of characters of the symmetric group $5. Hall-Littlewood-Macdonald symmetric polynomials 56. Duality $7. GHL-functions and generalized Legendre polynomials $8. Determinantal formulas for GHL-polynomials $9. Branching of Macdonald polynomials Chapter 3. The Plancherel Measure of the Symmetric Group $1. The typical shape of random Young diagrams $2. Gaussian limit for the Plancherel measure $3. Distribution of symmetry types of high rank tensors $4. A q-analogue of the hook walk algorithm $5. The q-analogue of the hook-length formula $6. Multiple Selberg integrals Chapter 4. Young Diagrams in Problen~sof Analysis $1. Rectangular diagrams and rational fractions $2. Continuous diagrams and R-functions $3. The Krein correspondence

vii

...

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CONTENTS

54. 55. 56. 57.

Interval shrinkage process Differential model of growth of Young diagrams Plancherel growth and semicircle diffusion Asymptotic separation of roots of orthogonal polynomials

References Comments on Kerov's Thesis. By G. OLSHANSKI Additional References

Foreword This book reproduces, without modifications, the "Doctor of Sciences" thesis of the remarkable mathematician Sergei Vassilievich Kerov (June 12, 1946-July 28, 2000). The thesis was written in 1992-1993 and defended in 1994. It is devoted mostly to analytical aspects of the asymptotic representation theory of symmetric groups and related problems of analysis. I have t o say several words about these topics, which appeared about 30 years ago. The theory itself and the name "Asymptotic representation theory" were suggested by me in the late sixties; this name refers to the whole complex of problems that lie at the interface of functional analysis, algebra, combinatorics, and probability theory, and concern the study of the behaviour of classical groups and their representations as the rank (degree) of the group tends to infinity. It is natural here to single out two classes of problems: proper asymptotic problems and problems that concern the limiting object, i.e., an infinite or infinitedimensional group which is an infinite analogue of a classical group. The principal example, which serves as a model for more difficult cases, is of course the asymptotic theory of symmetric groups and their representations. It was with this example that our activity started, just as in the 19th century the theory of finite groups and their representations had started with the symmetric groups. An early example of an asymptotic problem of the first kind was considered in the early seventies in my papers with A. Shmidt [20, 211' on the limiting joint distribution of the normalized cycle lengths of uniformly distributed random permutations. Combinatorics is here closely intertwined with probability theory. Later, many papers were devoted t o this topic, which has numerous applications (in number theory, coagulation schemes, populational genetics, etc.). This topic is considered partially in Chapter 4 of Kerov's thesis; it is also related to his later papers on the Poisson-Dirichlet measures. I formulated the dual asymptotic problem for symmetric groups in the early seventies, and it was solved in joint papers with Kerov [12, 181; this was the problem about the limiting shape of Young diagrams with the Plancherel statistics, or the typical representation of the symmetric group of high degree. From the technical point of view, it belongs to the class of "limit shape problems", which has recently become very popular. Namely, the problem asks what is the asymptotic behaviour of a configuration (for example, a Young diagram) that grows randomly but according t o certain rules. This topic is developed in Chapter 3 of the thesis, where, in particular, the central limit theorem for the Plancherel measure is proved, and numerous relations to other problems of analysis are considered. For further development see, e.g., [A.28, A.27, A.36, A.41, A.45, A.7, A.8, A.25, A.261. 'A reference such as [A.n] refers t o the additional reference list given a t the end of this volume. A reference without A refers t o the reference list in Kerov's thesis.

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FOREWORD

A survey of a more general class of limiting shape problems can be found in [A.52] and in the papers by A. Okounkov on Schur measures, e.g., [A.38]. It was clear from the very beginning that the problem concerning the asymptotic behaviour of representations and characters is of fundamental importance. However, only much later, in the nineties, did it become clear that this problem has many relations to other topics: spectra of random matrices (see, e.g., [A.50, A.3, A.4, A.29, A.30, A.15]), free probability theory (see, e.g., [A.5-A.lO, A.48, A.49, A.14]), integrable problems and algebraic geometry (see, e.g., [A.2, A.38, A.40, A.42, A.43]), the theory of z-measures (see, e.g., [A.13, A.391). This list is far from complete. Perhaps, one relation is worth special mention. I mean the solution of the so-called Ulam's problem on the longest increasing subsequence in a sample of n independent random variables uniformly distributed on an interval (or the longest increasing subsequence in a random permutation). This problem has a long history; an important role here is played by the RSK algorithm which interprets the length of this subsequence as the length of the first row of a random Young diagram. The problem was solved completely in our paper [12], where we proved the old conjecture that the answer is 2 6 . Note that the abovementioned theorem on the limit shape of a Young diagram, which was obtained in our work and, independently, by Logan and Shepp [138],and which gives complete information on the entire monotone structure of a random sample, is, however, not sufficient t o find the asymptotics of the length of the first row, since this theorem gives only a lower bound on this length. So it required an important, significant technical observation related to Young tableaux, which was made by Kerov and completed the solution of Ulam's problem. Our proof of the general limiting shape theorem was based on a continuous analogue of the hook-length formula, estimations of probabilities, and solution of an integral equation; it was different from the proof in [138] and allowed us also to obtain estimates for the asymptotics of the maximum dimension of representations of symmetric groups. For an elementary proof of Ulanl's problem, see, e.g., [A.l]. Problems of the second kind concern the study of the infinite analogue of symmetric groups, the infinite symmetric group of finite permutations. In this class of problenls, there is a distinguished problem concerning the description of the characters of this group, which was solved by Thoma [161]by analytical techniques and which was considered from the new point of view in our paper [1.3],and solved by means of a very general ergodic method suggested in [lo] (see the first chapter of the thesis). It can be naturally reformulated as a problem on the boundary of the Young graph. In fact, this is a new class of problems which can be stated in purely analytical terms - as the description of harmonic functions of certain operators; in probabilistic terms - as the description of the Poisson or Martin boundary of as the computation of traces of AF-algebras; random walks; in algebraic terms and, finally, in combinatorial terms - as the statistics of the number of paths and central measures in graded graphs. Although the ergodic method provides a general approach, analytical difficulties are different in each specific case, and they are far from easy to overcome. Our work on the computation of the characters of the infinite symmetric group by the ergodic method [I131 was continued in papers by Okounkov [54], Olshanski and Borodin [A.12], and others, where several new proofs of Thoma's theorem were suggested and a number of other graphs were considered. The leading role here belongs to Kerov; he considered different branching -

FOREWORD

ix

parameters (see Chapter 1). The results in this field that were known by 1993 are collected in Chapters 1 and 2. Later this problem was intensely studied by Kerov for other graphs and measures on them (Chapter 2). For further results about central measures and boundaries of different graphs, see, e.g., [A.24, A.21, A.22, A.23, A.19, A.201). In subsequent work, the ergodic method was repeatedly used for the description of invariant measures, characters, etc., see, e.g., [A.46, A.441. A very important general conjecture by Kerov (see [119]) on the boundary of graphs related to the Hall-Macdonald polynomials is still open. In a subsequent paper, written by Kerov together with Olshanski and Okounkov [A.34],the solution is found for a special case of Kerov's conjecture, namely, for the so-called Jack graph, a version of the Young graph with different transition probabilities. I think that the most interesting and original part of Kerov's thesis is the study of continuous Young diagrams. Already the first paper [12] on the limit shape of Young diagrams with the Plancherel distribution contained the transition from ordinary diagrams to continuous ones, and a continuous analogue of the hook-length formula. Kerov showed (see Chapter 4), first, that the limit shape of random Young diagrams with the Plancherel distribution appears naturally in a seemingly quite unrelated problem concerning separation of roots of orthogonal polynomials; and, which is even more important, that there exists a one-to-one correspondence between continuous diagrams and probability distributions that extends the correspondence between ordinary Young diagrams and their Plancherel transition probabilities; this correspondence is a nonlinear transformation of measures, which he rightly called the Markov-Krein transform. Thus Kerov linked the classical moment problem to the combinatorics of continuous Young diagrams. Ordinary diagrams correspond in this picture to discrete interlacing measures, and their Markov-Krein transform corresponds to the partial fraction expansion. Not less impressive is the relation, discovered by Kerov, of the Plancherel dynamics of continuous Young diagrams to a special solution of the Burgers equation: it turns out that the same limit shape of Young diagrams is a fixed point for this equation, and it attracts asymptotically the solutions of a certain class. And the most classical-looking result is the theorem that says that the interlacing roots of two adjacent orthogonal polynomials generate a Young diagram, and this diagram converges, as the number of the orthogonal polynomials goes t o infinity, t o the same limit shape. Kerov also found close relations of this topic to Voiculescu's free probability theory (the role of the Gaussian law in classical probability theory is played here by the senlicircle law which is related to the same limiting curve) and to combinatorics (the hook walk and the interval shrinkage process). For further results concerning the combinatorial aspects, see, e.g., [A.47, A.17, A.181. The results of Kerov on continuous diagrams and the Markov-Krein transform are set forth in more detail in his subsequent paper [A.33]. For further generalizations of the Markov-Krein transform, see [A.35, A.511. During the intervening years the results of the thesis did not become outdated; they are as interesting t o read now as they were at that time. Of course, the abovementioned progress in the study of more subtle asymptotics and relations to random matrices and free probability could not be mentioned in the thesis; however, Kerov took an active part in this new research (see the list of his publications, at the end of this Foreword).

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FOREWORD

The thesis is written very clearly and can be regarded as a handbook for mathematicians who want to work in asymptotic representation theory and obtain information on many results of the first period of its development. The introduction, which occupies almost a quarter of the whole thesis, contains all definitions and main results, so that a reader who is interested only in statements will need to read only this extensive introduction. The thesis includes only a part of the work done by Kerov up to 1994; the complete bibliography is given in this volume. I have asked G. Olshanski, A. Gnedin, and N. Tsilevich to prepare an additional list of references to the papers of subsequent years related to the topics touched upon in the thesis. Olshanski also prepared bibliographic comments, which are given at the end of the book. These comn~entscover only some of the topics that were studied in Kerov's thesis and developed later. The current state of this broad and rapidly developing area, asymptotic representation theory and its applications, demonstrates a very diverse and vivid picture. Without any doubt, Kerov's work will be avidly studied by future generations of mathematicians. As an addendum to this Foreword we reproduce an abridged version of the obituary which I wrote for the special volume "Representation theory, dynamical systems, combinatorial and algorithmic methods. VI", Zapiski Nauchn. Semin. POMI, v. 283, 2001, which was dedicated to the memory of S. Kerov; this obituary is translated by N. B. Lebedinskaya.

S. V. Kerov was a profound and original mathematician. He was gathering force not very rapidly, steadily advancing to more and more difficult problems and to the understanding of various mathematical connections. The list of his articles is not very large, but there are a number of serious papers, which will be studied and continued. His absolutely unexpected and sudden death did not allow him to complete several articles that were almost finished. I was his scientific adviser when he wrote his graduation thesis and when he worked at his Candidate's (Ph.D.) thesis.* For many years, he was my colleague and coauthor; together we elaborated ideas that I had worked on earlier, as well as those that occurred to both of us later. At a later time, he became a prominent expert on combinatorics, the theory of symmetric functions, and many other fields. He carefully examined the literature, and I often learned new information from him. His own ideas and papers, especially in the last ten years, showed his deep insight into combinatorics, analysis, and probability. Here I will try to outline his scientific path - I was a witness of much of it. In particular, I discuss in more detail our joint work, which began in the mid-seventies, continued with interruptions until the mid-eighties, and then recommenced in the late nineties. From his third year at Leningrad University, Sergei participated in the seminar on dynamical systems, representations, and algebras that I have headed since 1967. His M. Sci. thesis was devoted to flows on nilpotent and solvable manifolds. *Translatzon Editor's Note. The Russian Candidate's thesis is equivalent t o the Western Ph.D. thesis. The Russian Doctoral thesis, which this book is, is much more advanced, roughly equivalent t o "first published book".

FOREWORD

xi

Sergei was interested in different areas of mathematics, and from the very beginning he was very earnest and thorough in his work and study. There were many talented students (Ya. Eliashberg, Yu. Matiyasevich, and others) of the same year as Sergei, but even at that time he had a high reputation, which was based on the impression of his unceasing internal activity. His modest and dignified manner was very attractive. In 1969, he became a post-graduate student at the Department of Analysis, and I was his scientific adviser. In 1969-70, I gave a course of lectures on C*-algebras and related topics. It was a new topic to me, and I wanted to apply these techniques t o dynamical systems and representation theory. Since representation theory had gradually become the main topic, the seminar was guided in major part by problems in this theory. Sergei studied the theory of C*-algebras and the classical theory of representations of symmetric and finite groups. As a subject for his Candidate's thesis, I suggested the duality theory of *-algebras (the theory of "positive" duality). I put forward this idea generalizing the theory of Hopf algebras in 1971 and published a brief note on the geometry of states (the theory of packets) in 1972. The basic definition implied that for algebras in linear duality, niultiplication in each of them (or multiplication and comultiplication) is ail operation preserving positivity (but not multiplicativity as in Hopf algebras). The main problems were formulated, and Sergei tackled them with enthusiasm. In his Candidate's thesis, he carefully examined the finite-dimensional version, including the Plancherel duality and induction; he also studied the nontrivial commutative version (see [A.31, A.321). This theory is not yet entirely known, but Kerov's two papers on this subject are often cited. Only recently, in papers with I. Ponomarenko and S. Evdokimov, we proved that the finite-dimensional algebras in Plancherel duality are just the so-called C-algebras in algebraic combinatorics. I have no doubt that these ideas will be used in the future. There are also many relations with hypergroups, the theory of generalized shifts, quantum groups, many-valued groups, and so on. Sergei studied B. M. Levitan's book thoroughly, and we discussed further applications of the theory of positive duality to differential equations. By 1975, when his Ph.D. thesis was completed, I had drawn him t o a new topic, asymptotic representation theory. We started by studying Thoma's work [161] on the characters of the infinite symmetric group (I learned about this paper from I. M. Gelfand and I. Segal) and with the problem concerning the asymptotics of the Plancherel measure on Young diagrams, which seemed to me t o be of key importance. An important role in investigating these problems was played by our experience in ergodic theory and dynamical systems (ergodic method, invariant central measures, etc.). We worked a lot with AF-algebras and the K-theory of these algebras, and rediscovered some known facts. I think that our most important result in this field is the computation of the K-functor of the group algebra of the symmetric group [128]. Sergei's main research interest remained, however, related t o questions concerning Young diagrams, symmetric groups, and related problems of analysis and combinatorics. These interests are well reflected in the present book. Our plans included the study of asymptotic problems concerning representations of matrix groups over finite fields: this topic was outlined in the eighties, and we discussed it with A. Zelevinsky, who worked a t that time on the application of the method of Hopf algebras to a somewhat different problem about the structure of the set of representations of all finite groups GL(n, F,). However, it was not until 1996 that we started a serious study of the characters of the groups G L ( m , F,) and

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Publications of Sergei Kerov K.1. Double function algebras on a finite group, Zap. Nauchn. Sem. LOMI 39 (1974), 182-185; English transl., J . Soviet Math. 8 (1977), 136-139. K.2. Duality of finite-dimensional *-algebras, Vestnik Leningrad. Univ. 1974,no. 7 (Ser. Mat. Mekh. Astr. vyp. 2 ) , 23-29; English transl., Vestnik Leningrad Univ. Math. 7 (1979), 122130. K.3. Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux ( w i t h A . M. Vershik), Dokl. Akad. Nauk S S S R 233 (1977), 1024-1027; English transl., Soviet Math. Dokl. 18 (1977),527-531. K.4. Characters and factor representations of the i n j n i t e symmetric group ( w i t h A . M. Vershik), Dokl. Akad. Nauk S S S R 257 (1981), 1037-1040; English transl., Soviet Math. Dokl. 23 (1981), 389-392. K.5. Asymptotic theory of the characters of a symmetric group ( w i t h A . M . Vershik), Funktsional. Anal. i Prilozhen. 15 (1981), no. 4 , 15-27; English transl., Funct. Anal. Appl. 15 (1981), 246-255. K.6. Characters and factor-representations of the infinite unitary group ( w i t h A . M. Vershik), Dokl. Akad. Nauk S S S R 267 (1982),272-276; English transl., Soviet Math. Dokl. 26 (1982), 570-574. K.7. Polynomial dimension groups, Operator Theory and Function Theory, Vol. I ( B . S. Pavlov, editor), Leningrad Univ., Leningrad, 1983, pp. 185-194. (Russian) K.8. The K-functor (Grothendieck group) of the infinite symmetric group ( w i t h A . M. Vershik), Zap. Nauchn. Sem. LOMI 123 (1983), 126-151; English transl., J . Soviet Math. 28 (1985), 549-568. K.9. W-graphs of representations of symmetric groups, Zap. Nauchn. Sem. LOMI 123 (1983), 190-202; English transl., J . Soviet Math. 28 (1985), 596-605. K.lO. Experiments i n calculating the dimension of a typical representation of the symmetric group (with A . Vershik and A. Gribov), Zap. Nauchn. Sem. LOMI 123 (1983), 152-154; English transl., J . Soviet Math. 28 (1985), 568-570. K . l l . Isomorphisms of rings constructed by simplicia1 schemes ( w i t h A . Karp), Questions o f Differential Geometry "In t h e Large" ( A . L. Verner, editor), Leningrad Gos. Ped. Inst., Leningrad, 1983, pp. 60-66. (Russian) K.12. Characters, factor representations and K-functor of the infinite symmetric group ( w i t h A . M. Vershik), Operator Algebras and Group Representations, Vol. I1 (Proc. Internat. Conf., Neptun, 1980; Gr. Arsene et al., editors), Monogr. Stud. Math., vol. 18, Pitman, Boston, M A , 1984, pp. 23-32. K.13. The Robinson-Schensted-Knuth correspondence and the Littlewood-Richardson rule, Uspekhi Mat. Nauk 39 (1984), no. 2, 161-162; English transl., Russian Math. Surveys 39 (1984), no. 2, 165-166. K.14. Asymptotic behavior of the m a x i m u m and generic dimensions of irreducible representations of the symmetric group ( w i t h A . M. Vershik), Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 25-36; English transl., Funct. Anal. Appl. 19 (1985), 21-31. K.15. Locally semisimple algebras. Combinatorial theory and the KO-functor ( w i t h A. M. Vershik), Itogi Nauki i Tekhniki: Sovremennye Problemy Mat., vol. 26, V I N I T I , Moscow, 1985, pp. 3-56; English transl., J . Soviet Math. 38 (1987), 1701-1733. K.16. The characters of the infinite symmetric group and probability properties of the RobinsonSchensted-Knuth algorithm ( w i t h A . M . Vershik), SIAM J . Algebraic Discrete Methods 7 (1986), 116-124.

xiv

PUBLICATIONS O F SERGE1 KEROV

K.17. Combinatorics, the Bethe ansatz and representations of the symmetric group ( w i t h A. Kirillov and N. Reshetikhin), Zap. Nauchn. Sem. LOMI 155 (1986), 50-64; English transl., J . Soviet Math. 41 (1988), 916-924. K.18. Distribution of symmetry types of high rank tensors, Zap. Nauchn. Sem. LOMI 155 (1986), 181-186; English transl., J . Soviet Math. 41 (1988), 995-999. K.19. Random Young tableaux, Teor. Veroyatnost. i Primenen. 31 (1986), 627-628; English transl., Theory Probab. Appl. 31 (1986),553-554. K.20. Realizations of *-representations of Hecke algebras, and Young's orthogonal form, Zap. Nauchn. Sem. LOMI 161 (1987), 155-172; English transl., J . Soviet Math. 46 (1989), 2148-2158. K.21. Realzzations of representations of the Brauer semigroup, Zap. Nauchn. Sem. LOMI 164 (1987), 189-193; English transl., J. Soviet Math. 47 (1989), 2503-2507. K.22. Characters and realizations of representations of the infinite-dimenszonal Hecke algebra, and knot invariants ( w i t h A . M. Vershik), Dokl. Akad. Nauk S S S R 301 (1988), 777-780; English transl., Soviet Math. Dokl. 38 (1989), 134-137. K.23. AF-algebras of truncated Pascal triangles ( w i t h V . Volchegursky), Problems in Group Theory and Homological Algebra ( A . L. Onishchik, editor), Yaroslav. Gos. Univ., Yaroslavl, 1989, pp. 29-37. (Russian) K.24. Combinatorial examples i n the theory of AF-algebras, Zap. Nauchn. Sem. LOMI 172 (1989), 55-67; English transl., J . Soviet Math. 59 (1992), 1063-1071. Zap. K.25. A n algebra of invariants for the action of the group S p ( 2 m ) i n the algebra @M2mC, Nauchn. Sem. LOMI 172 (1989), 68-77; English transl., J. Soviet Math. 59 (1992), 10721078. K.26. Representattons, maximal with respect to dimension, of symmetric groups (with A . Pass), Zap. Nauchn. Sem. LOMI 172 (1989), 160-166; English transl., J . Soviet Math. 59 (1992), 1131-1135. K.27. T h e Grothendieck group of the infinite symmetric group and symmetric functions (with the elements of the theory of KO-functor of AF-algebras) ( w i t h A . M. Vershik),Representation o f Lie Groups and Related Topics ( A . M . Vershik and D. P. Zhelobenko, editors), Adv. Stud. Contemp. Math., vol. 7 , Gordon and Breach, New Y o r k , 1990, pp. 36-114. K.28. Random processes with common cotransition probabilities ( w i t h 0. Orevkova), Zap. Nauchn. Sem. LOMI 184 (1990), 169-181; English transl., J . Math. Sci. (New Y o r k ) 68 (1994), 516-525. K.29. Hall-Littlewood functions and orthogonal polynomials, Funktsional. Anal. i Prilozhen. 25 (1991),no. 1, 78-81; English transl., Funct. Anal. Appl. 25 (1991), 65-66. K.30. Characters of Hecke and B i m a n - W e n z l algebras, Quantum Groups (Leningrad, 1990), Lecture Notes Math., vol. 1510, Springer-Verlag, Berlin, 1992, pp. 335-340. K.31. Trzangularity of transition matrices for generalized Hall-Littlewood polynomials, Quantum Groups (Leningrad, 1990), Lecture Notes Math., vol. 1510, Springer-Verlag, Berlin, 1992, pp. 389-390. K.32. Generalized Hall-Littlewood symmetric functions and orthogonal polynomials, Representation Theory and Dynamical Systems, A d v . Soviet Math., vol. 9 , Amer. Math. Soc., Providence, RI, 1992, pp. 67-94. K.33. q-analog of the hook walk algonthm and random Young tableaux, Funktsional. Anal. i Prilozhen. 26 (1992),no. 3 , 35-45; English transl., Funct. Anal. Appl. 26 (1992), 179-187. K.34. Combinatorics of rational representations of the group G L ( n , @ )( w i t h A . N . Kirillov), Zap. Nauchn. Sem. POMI 200 (1992),83-90; English transl., J . Math. Sci. (New Y o r k ) 77 (1995), 3190-3194. K.35. Gaussian limit for the Plancherel measure of the symmetric group, C . R . Acad. Sci. Paris S6r. I Math. 316 (1993), 303-308. K.36. A q-analog of the hook walk algorithm for random Young tableaux, J . Algebraic Combin. 2 (1993), 383-396. K.37. Harmonic analysis on the infinite symmetric group ( w i t h G . I . Olshanski and A. M. Vershik), C . R . Acad. Sci. Paris S6r. I Math. 316 (1993), 773-778. K.38. Asymptotics of the separation of roots of orthogonal polynomials, Algebra i Analiz 5 (1993), no. 5 , 68-86; English transl., St. Petersburg Math. J . 5 (1994), 925-941.

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K.39. Transition probabilities of continual Young diagrams and Markov m o m e n t problem, Funktsional. Anal. i Prilozhen. 27 (1993), no. 2, 32-49; English transl., Funct. Anal. Appl. 27 (1993), 104-117. K.40. The Plancherel growth of Young diagrams and the asymptotics of interlacing sequences, Dokl. Akad. Nauk 333 (1993), 8-10; English transl., Russian Acad. Sci. Dokl. M a t h . 48 (1994);420-424. K.41. The asymptotics of interlacing sequences and the growth of continuous diagrams, Zap. Nauchn. S e m . POMI 205 (1993), 21-29; English transl.. J . Math. Sci. ( N e w Y o r k ) 80 (1996), 1760-1767. K.42. Polynomial functions o n the set of Young diagrams ( w i t h G . I . Olshanski), C . R . Acad. Sci. Paris S8r. I Math. 319 (1994), 121-126. K.43. Asymptotics of large random Young diagrams, Abstracts S i x t h C o n f . Formal Power Series and Algebraic Combinatorics, D I M A C S , 1994, pp. 285-294. K.44. Asymptotic representation theory of the symmetric group and its applications i n analysis, D. Sci. thesis, Steklov Institute o f Mathematics at S t . Petersburg, St. Petersburg, 1994; English transl., this volume. K.45. Stick breaking process generates virtual permutations with the Ewens distribution ( w i t h N . V . Tsilevich), Zap. Nauchn. S e m . POMI 223 (1995), 162-180; English transl., J . M a t h . Sci. ( N e w Y o r k ) 87 (1997), 4082-4093. K.46. Subordinators and permutation actions with quasi-invariant measure, Zap. Nauchn. S e m . POMI 223 (1995), 181-218; English transl., J. Math. Sci. ( N e w Y o r k ) 87 ( 1 9 9 7 ) ,4094-4117. K.47. Coherent allocations; and the Ewens-Pitman formula, Preprint POMI 21/1995 ( 1 9 9 5 ) , 1-15. K.48. Small cycles of big permutations, Preprint POMI 22/1995 ( 1 9 9 5 ) , 1-9. K.49. A dzflerential model for the growth of Young diagrams, T r u d y Sankt-Peterburg. Mat. Obshch. IV (1996), 165-192; English transl., Amer. Math. Soc. Transl. ( 2 ) 188 (1998), 111-130. K.50. The boundary of Young lattice and random Young tableaux, Formal Power Series and Algebraic Combinatorics ( N e w Brunswick, N J , 1994), D I M A C S Ser. Discrete M a t h . Theoret. C o m p u t . Sci., vol. 24, Amer. Math. Soc., Providence, RI, 1996, pp. 133-158. K.51. Rook placements o n Ferrers boards, and.matrix integrals, Zap. Nauchn. S e m . POhtI 240 (1997), 136-146; English transl., J . Math. Sci. ( N e w Y o r k ) 96 (1999), 3531-3536. K.52. Interlacing measures, Kirillov's Seminar o n Representation Theory, Amer. Math. Soc. Transl. ( 2 ) 181 (1998), 35-83. K.53. T h e boundary of the Young graph with Jack edge multiplicities ( w i t h A . Okounkov and G . Olshanskii), Internat. M a t h . Res. Notices 1998,173-199. K.54. O n a group of infinite-dimensional matrices over a finite field ( w i t h A . M. V e r s h i k ) , Funktsional. Anal. i Prilozhen. 32 (1998), no. 3 , 3-10; English transl., Funct. Anal. Appl. 32 (1999), 147-152. K.55. T h e algebra of conjugacy classes i n symmetric groups, and partial permutations ( w i t h V . Ivanov), Zap. Nauchn. S e m . POMI 256 (1999), 95-120; English transl., J. M a t h . Sci. ( N e w Y o r k ) 107 (2001), 4212-4230. K.56. Anisotropic Young diagrams and Jack symmetric functions, Funktsional. Anal. i Prilozhen. 34 (2000),no. 1 , 51-64; English transl., Funct. Anal. Appl. 34 (2000), 41-51. K.57. The Plancherel measure of the Young-Fibonacci graph ( w i t h A . G n e d i n ) , Math. Proc. C a m bridge Philos. Soc. 129 (2000), 433-446. K.58. The Martin boundary of the Young-Fibonacci lattice ( w i t h F. G o o d m a n ) , J. Algebraic Combin. 11 (2000), 17-48. K.59. E q u i l i b ~ u mand orthogonal polynomials, Algebra i Analiz 12 (2000),no. 6 , 224-237; English transl., S t . Petersburg Math. J . 12 (2001), 1049-1059. K.60. The Markov-Krein correspondence i n several dimensions ( w i t h N . V . Tsilevich), Zap. Nauchn. Sem. POMI 283 (2001), 98-122; English transl., t o appear i n J. Math. Sci. ( N e w York). K.61. A characterization of G E M distributions ( w i t h A.Gnedin), Combin. Probab. C o m p u t . 10 ( 2 0 0 l ) , 213-217. K.62. Fibonacci solitaire ( w i t h A . Gnedin), R a n d o m Structures Algorithms 20 ( 2 0 0 2 ) , 71-88.

CHAPTER 0

Introduction 0. Plan of the book

The first main subject of this work is the description of asymptotic properties of the characters of finite symmetric groups and the related construction of the characters of the infinite symmetric group. More generally, the combinatorial techniques developed here can be applied to a wide class of algebras that can be approximated by finite-dimensional semisimple algebras, in particular, to the group algebras of locally finite groups. The problem of computing the characters of such groups and algebras is reduced to a typical problem of potential theory on countable graphs, namely, to the description of nonnegative harmonic functions on them. Similarly to the classical situation of the unit disk, the nonnegative harmonic functions are constructed by the (generalized) Poisson integral taken over the ideal boundary of the graph. The computation of this boundary, which is similar to the classical Martin boundary, is usually a difficult problem. In this book we carry out this computation for a number of examples of combinatorial and algebraic origin related to the lattice of Young diagrams. Among all characters of the infinite symmetric group, we are most interested in the character of its regular representation. We give a detailed description of the shape of large Young diagrams typical with respect to the Plancherel measure on the finite symmetric group, and also study stochastic properties of the deviations of random diagrams from the limiting curve. In other terms, we investigate the limits of joint distributions of characters of irreducible representations of the symmetric group of degree n with respect to its Plancherel measure. The asymptotic analysis of the Plancherel measure is the second important component of this book. The third main subject is related to continuous limits of numerous nontrivial combinatorial constructions and algorithms of the theory of symmetric groups and their representations. We establish a direct relation between the transition probabilit,ies of the Plancherel measure of the infinite symmetric group and the partial fraction expansions of rational functions with interlacing zeros and poles. Using this relation, we manage to obtain a continuous analogue of the hook walk algorithm, which is well-known in the combinatorics of Young diagrams. This construction in turn provides a completely new description of the relation between the classical moment problems of Hausdorff and Markov. In probabilistic terms, the limiting algorithm (we call it the interval shrinkage algorithm) allows us to construct the distributions of integrals over random Dirichlet measures. The dynamical process of random growth of Young diagrams, which we regard as the Plancherel measure of the infinite symmetric group, leads in a natural way to a continuous model which is equivalent to the first order partial differential equation

Rt+RR,=O.

2

0.

INTRODUCTION

The typical limit shape of large random Young diagrams (we call it the arcsine law) arises unexpectedly often in seemingly unrelated problems. We give two examples of such situations: the asymptotic behaviour of interlacing roots of orthogonal polynomials and the perturbation of spectra of typical large random matrices under a linear constraint. Within these three areas (the asymptotic theory of characters of the infinite symmetric group, limiting properties of the Plancherel measure of this group, and applications of the representation theory of symmetric groups in analysis), we discuss a q-analogue of the hook walk algorithm, study in detail Macdonald symmetric polynomials, and compute the boundaries of the related deformations of the Young graph which is the central object of all our considerations. This book consists of a detailed introduction and four chapters subdivided into sections. We start with a preliminary list of the results and necessary references. A more detailed description of the contents is given below in the main part of the introduction. In the first chapter we present an account of general methods for dealing with algebras that have an approximation by finite-dimensional semisimple subalgebras (we call them locally semisimple, or, for short, LS-algebras). Conceptually, the theory of such algebras goes back to von Neumann, and the simplest examples were considered in the 60s by Glimm and Dixmier. A systematic study of these algebras was initiated by Bratteli's paper [74], which laid the foundations of the combinatorial approach t o LS-algebras and his subsequent papers [75, 76, 771. A fundamental contribution to the theory was made a little later by Elliott [88],who showed that the Grothendieck group K Oendowed with a natural ordering structure is a complete invariant of an LS-algebra. Another key fact was discovered by Effros, Handelman, and Shen [87]; they found an abstract characterization of the class of all ordered abelian groups arising as Ko(A) for some LS-algebra A. A summary of the first ten years of the theory of LS-algebras is given in the survey [86]. A new impulse for the study of LS-algebras was given by considering infinitedimensional analogues of classical groups such as the unitary group

(see [160]) and the infinite symmetric group

These examples, which are far more difficult than those studied at the initial stage of the theory, considerably influenced the development of its general techniques. The attention of mathematicians became focused on the problem of computing the characters of groups that have an approximation by finite or compact subgroups. A substantially new approach to the analysis of this problem consisted in regarding LS-algebras as crossed products associated with dynamical systems. The ergodic method for constructing characters, suggested in [13],is based on this approach, and develops A. M. Vershik's ideas applied earlier in [ l o ] . Let us also mention two other new results from the general theory of LSalgebras, which were discovered in [13] and [126] when working with the group 6,. The first one establishes the key role of the notion of semzfinite characters of LS-algebras, introduced in [13],in the description of positive elements of the

0. P L A N OF T H E BOOK

3

Grothendieck group KO. The second one consists in introducing the class of multiplicative Bratteli diagrams and developing the related theory of ordered Riesz rings. A survey of these results can be found in papers by A. M. Vershik and the author [16], [17], [128]. Continuous analogues of branching graphs and related problems are considered in [42]. The central problem of the second chapter is the approximative computation of the characters of the infinite symmetric group 6,. The characters and representations of the finite symmetric groups were found at the beginning of the twentieth century by F'robenius [61] and Scliur [155]. Necessary combinatorial techniques were developed in a long series of papers by Young; the list of these papers can be found in the last paper of the series [168]. In Weyl's famous book [8]the representations of the symmetric groups are closely intertwined with the representations of the classical matrix groups. Among the great number of textbooks and monographs treating the theory of characters and representations of B,, we would like to mention [26, 29, 50, 621, [113, 136, 1521. The monograph [81]is devoted to probabilistic problems on the group of permutations. A combinatorial basis for the representation theory of symmetric groups is provided by Young diagrams and Young tableaux. We borrow the terminology and necessary background on the algebra of symmetric polynomials from Macdonald's remarkable monograph [49]. It is worth noticing that, in spite of its hundredyear history, the combinatorics of permutations and their representations is still actively developing. We mention a combinatorial version of the Fourier transform for 6, [152], [157], [133],the hook-length formula [92] for the dimensions of its irreducible representations, various descriptions of the Littlewood-Richardson rule for the decompositions of tensor products of representations of unitary groups, the hook walk algorithm [102], [103], and a relation to the Bethe ansatz [41] (the list is not intended t o be exhaustive). The role of symmetric groups and Hecke algebras in Jones' recent construction of new topological invariants of knots and links is not less impressive, see [114], [19]. The characters of the infinite symmetric group 6 , were first found by Thoma [161], his result being surprisingly equivalent to a purely analytical problem of characterizing totally positive sequences and series which was solved independently by Schoenberg and his colleagues [64]. A completely different description of the characters of 6, obtained in [13]is a part of a wider project of Vershik for the study of approximation problems. For other realizations of Vershik's approach see, for example, his papers [20], [119],

[lo].

The problem of computing the characters of 6, is equivalent to the problem of finding nonnegative harmonic functions on the lattice (graph) of Young diagrams. In the second chapter we present a detailed study of the similar problem for a certain two-parameter deformation of the Young graph which reflects the branching of Macdonald symmetric polynomials introduced in [139], [140]. For appropriate values of the parameters of this deformation we recover the results of Kingman [129], [130] and Nazarov [51], and also obtain the Jack graphs which are used in the third chapter for computing multivariate Selberg integrals. This part of the book is based on the author's papers [34], [35],[119]. One of the first examples of describing harmonic functions on graphs is provided by de Finetti's classical theorem on random sequences invariant under finite permutations. Another important source of discrete problems on positive harmonic

4

0.

INTRODUCTION

functions is the theory of random walks on groups; see the survey by Kaimanovich and Vershik [115] for an introduction to this theory. In the third chapter we study in detail the most interesting character of the infinite symmetric group, the character of its regular representation. First we introduce the general notion of the Plancherel measure as a Markov chain on the branching graph of irreducible representations of approximating finite groups; this definition is convenient for working with arbitrary locally finite groups. Then we reproduce the old theorem of Vershik and the author [12], [la]on the asymptotics of the shape of large random Young diagrams ( a similar fact was obtained independently in [138]). Most of the rest of the results discussed later in this book are motivated by this theorem. The new results of the third chapter include the description of typical symmetries of high rank tensors [31],an analogue of the central limit theorem for the characters of the symmetric group 6, [40], and a q-analogue of the hook walk algorithm [36]. We also give a new derivation of multivariate Selberg integrals (see [156], [69], [151],[142]),based on the results of [46]. The fourth and final chapter contains the author's results related to applications of the asymptotic theory of characters of 6 , in analysis. Though they do not depend formally on the first three chapters and are of value in themselves, nevertheless the ideology and problems of the fourth chapter are motivated by the results on the Plancherel measure of 6, and the limit shape of large Young diagrams. I-t seems that there is a deep connection between the symmetric groups and the theory of functions, and our results, as well as the equivalence of the EdreiSchoenberg theorem and the theorem on the characters of G,, are manifestations of this connection. Indirect evidence of this is the work of Issai Schur, who obtained remarkable theorems in both fields. Our first result in the fourth chapter is based on the representation of pairs of interlacing sequences of real numbers by so-called "rectangular diagrams", which are generalizations of Young diagrams. This simple technique allows us to state the problem of the asymptotic behaviour of such pairs. Following [38],we prove that the limit shape of diagrams which describe the interlacing roots of classical orthogonal polynomials is the same as for large random Young diagrams. The second main result published in [37] establishes a bijection between diagrams and probability measures. Essentially the same correspondence, but obtained from completely different reasons, appears in Krein and Nudelman's book [47] in connection with the Markov moment problem, and in [82] it was used in the problem of computing the distributions of integrals over the random Dirichlet measures. The relation, discovered in [37],between the Krein correspondence and the Plancherel measure of the symmetric group 6, offers a tempting opportunity t o apply various combinatorial techniques of the theory of permutations and their representations to the moment problem and other analytical problems. Using this idea and passing appropriately to the limit in the hook walk algorithm, we obtain a new stochastic description of the Krein correspondence, the interval shrinkage process. The third important result (see [122], [40]) is the construction of a differential model of growth of Young diagrams which turns out to be equivalent to the wellstudied Burgers equation. We prove that the asymptotic behaviour established earlier for random Young diagrams holds also for this model. The relation of the

51. GENERAL THEORY O F LOCALLY SEMISIMPLE ALGEBRAS

5

differential model t o the semicircle diffusion in the free probability theory developed by Voiculescu and his pupils [166], [167], [71] is worth further studying. After these short explanations we proceed t o a more detailed exposition of our results.

$1. General theory of locally semisimple algebras The main object of our study is the infinite symmetric group 6,.

By definition,

6, is a discrete countable group which consists of finite permutations of positive integers (that is, of permutations for which almost all elements are fixed points). Denote by 6, the finite subgroup of permutations from 6, that leave the elements n + l , n+2,. . . unchanged; then the group 6, = 6, is a union of an increasing sequence of finite subgroups. Infinite groups with this property are called locally finite. Since the 70s, Bratteli, Elliott, and others have developed powerful combinatorial techniques for working with group algebras of locally finite groups. Recall the basic results of their theory. 1.1. Combinatorial theory of LS-algebras. A locally semisimple algebra (LS-algebra for short) is the inductive limit

of an inductive family

of finite-dimensional semisimple algebras U,. We will assume that all homomorphisms preserve unity. According to Wedderburn's theorem, each finite-dimensional semisimple algebra is isomorphic to a direct sum of full matrix algebras:

Here I?, is a finite set whose elements index the equivalence classes of complex irreducible representations nx of the algebra Q,, and d(X) = dim nx is the dimension of the irreducible representation corresponding to X E I?,. A finite-dimensional seniisimple algebra 8, is determined up t o isomorphism by the list of dimensions {d(X))xEr, of its irreducible representations. An embedding of finite-dimensional semisimple algebras

can be adequately described by a bipartite graph with vertex set Tn-1 U I?,. By and A E I?, are joined by an edge of multiplicity definition, the vertices X E x(X,A), where x(X,A) is the multiplicity of xx in the decomposition

We denote by n~ the irreducible representation of the algebra U, corresponding to A E .,?I By F'robenius reciprocity, the same multiplicities arise in the decomposition

0.

6

INTRODUCTION

of the induced representation

If the bipartite graphs of two homomorphisms i n ,j,: + 2, coincide, then there exists an invertible element b E U, such that j,(a) = bi,(a) b-' for all a E 24-1. It follows that an inductive family (1.1) of finite-dimensional semisimple algebras generates an infinite graph, called the branching graph (= Bratteli diagram) of this family. By definition, I' is the union of the bipartite graphs associated with the embeddings in: UnP1 + U,. The vertex set of I' is I' = UTz0r n .

FIGURE 1. The Young graph Our most important example is related t o the inductive family

of the group algebras of the finite symmetric groups 6,. For convenience, we introduce the group algebra @[eO] -. @ of the nonexisting group Go; we identify it with The branching graph of this family is the Young graph shown in the algebra @[el]. Figure 1. Its vertices are Young diagrams, and the edges connect pairs of diagrams which differ from each other by a single square*) . The Young graph describes the branching of irreducible representations of the symmetric groups under restricting *) We write X

/* A if a Young diagram A is obtained from a diagram X by adding one square.

6 1 . GENERAL THEORY O F LOCALLY SEMISIMPLE ALGEBRAS

7

to a subgroup, and inducing

Note that all edges of the Young graph are simple. 1.2. Characters of LS-algebras and harmonic functions on graphs. The branching graph contains exhaustive information on an inductive family of finite-dimensional semisimple algebras, and hence on the LS-algebra U, = 1 4 '21, which is the inductive limit of this family. In particular, the characters of the algebra U, can be described in terms of the branching graph. A character of an LS-algebra U, is a linear functional

that enjoys the following three properties: (a) (positive definiteness) yi(aa*) 0 for all a E a,*) ; (b) (centrality) +(ab) = yi(ba) for all a , b E U,; (c) (nornlalization) $(I) = 1. In particular, the restriction of a character of the algebra U, gebra Un can be written as

>

+

to the subal-

Comparing with the decomposition

we conclude that the coefficients p(X) satisfy the condition

for all X E r,-l. DEFINITION. Let y:I? 4 @ be a function on the vertex set of a branching graph I?. We call it harmonic if (2.2) holds for all n = 1 , 2 , . . . . In our considerations, fornzula (2.2) plays the role of the mean value theorem for ordinary harmonic functions. The following simple result establishes a bijection between the characters of an LS-algebra and the harmonic functions on its branching graph. *) Here a H a* stands for a n arbitrary involution in Q, which is compatible with the standard involutions on the matrix subalgebras U ' , n = 1 , 2 , . . . . Although such a n involution is not unique, condition (a) does not depend on its choice.

8

0.

INTRODUCTION

THEOREM 1 ([17],59, Theorem 5). For each nonnegative harmonic function cp: r + R+ normalized so that 4 8 ) = 1 *), there exists a unique character $: U ', + C of the corresponding LS-algebra such that

for all n = 0 , 1 , 2 , . . . . Fornula (2.1) establishes a bijective homeomorphism between the space of characters of the LS-algebra Char('U,) and the space Harm(r) of normalized nonnegative harmonic functions on its branching graph. 1.3. The boundary of a branching graph. The space of harmonic functions Harm(r) endowed with the pointwise convergence topology is convex, compact, and metrizable. By Choquet's theorem (see, for example, [59]),each function cp E Harm(F) can be represented as

where M is a probability measure supported by the set of extreme points &(I?)= ex Harm(r). In fact, decomposition (3.1) with this property is unique; that is, the convex compact set Harm(r) is a simplex. This follows, for example, from Effros' results ( [ 8 6 ] ,Lemma 4.3). In all examples considered in this paper the set of extreme points &(I?)is closed in the simplex Harm(r). We call it the boundary of the branching graph r . Formula (3.1) allows us to identify the simplex of nonnegative harmonic functions Harm(r) with the set of all Bore1 probability measures on the boundary &(I?)of the branching graph. According to a classical theorem (see, e.g., [24], Chapter 1, Theorem 3.5), a nonnegative harmonic function on the disk has a unique Poisson integral representation

and the representing measure p can be obtained as the radial limit dQ p(d0) = lim cp(reie) r-1 2~

In Chapter 1 we establish similar facts for branching graphs. Towards this end, we define a generalized Poisson kernel @(X,6),where X E r, 6 E E ( r ) , and the radial embedding i: r --, E(I'), which possess the following properties: (a) For a fixed 6 E E ( r ) , the function cp(.) = a(.,6) lies in the simplex Harm(r). (b) The functions cPx(.) = @(A, .) are continuous on the boundary E ( r ) and distinguish the points of E ( r ) . (c) Given 6 E E, denote by Mn the probability distribution on the nth level rn of the branching graph I? with weights

*)We denote by 0 the unique point of the set

ro.

s l . GENERAL T H E O R Y O F LOCALLY SEMISIMPLE ALGEBRAS

9

where d(X) stands for the rl~imberof directed paths corlnecting Q) and A. Then the distributions i(hl,) weakly converge in & to the unit weight at tlie point 6 E &. Under these assumptions the following theorem holds. THEOREM 2 [123]. Conditions (a)-(c) i m p l y ( I ) each harmonic function cp E Harm(r) has a unique Poisson integral representation

uih,ere /L is a Borel probability measure o n th,e boundary &(I?); and (2) the representing measure p i s the weak limit of the discrete probability distributions i(ll.fn), where the measure M,,, i s determined by the h a r m o n i c function cp by form,ula (3.4). 1.4. C e n t r a l measures. Historically, one of the first problems reducible to compliting harmonic fiinctions on a graph was the problem of describing all Borel probability measures, on the space of 0- 1 sequences, that are invariant under finite permutatiorls of coordinates. The prod~ictmeasures &Ip, with a common probability p E [0, 11 of the unity for all coordinates, are examples of such measures. According to de Finetti's theorem (see [58],Vol. 2, Chapter 7, §4), each invariant measure M is a mixture of the product measlires:

where p is a urliqliely determined probability distriblition on the unit interval. Let us identify 0 - 1 sequences with paths in the Pascal triangle (see Figure 2). Then the invariance of M is eqliivalerlt to the following condition: ( C ) For any two paths u, I J of the graph I? from the initial vertex 0 to a common vertex X E I?, the hl-probabilities of u and v coincide and depend only on A. DEFINITION. A Borel probability measure M on the path space of a branching graph I? is said to be central if is satisfies condition (C). Denote by cp(X) the common probability (from corldition (C)) of all paths ending at X E I?. It is easy to check that cp is a harmonic function. This correspondence establishes a bijection between the space Cent(r) of central measures and Harm(r). Let 11sdenote by d ( u , A) the number of directed paths in I? from u E I? to X E r, and call it the dim,ension of the interval [u, A]. If u = 8 is the initial vertex of the graph, then we write simply d(X) = d(0, A). Given a fixed positive integer n, associate with a harmonic furlctiorl cp E Harm(r) the probability distribution on r,,,deiined by

The family of distributions {Mn),"==, is coherent in the following sense:

0.

INTRODUCTION

FIGURE 2. The Pascal triangle. Coherent families provide a convenient language for dealing with central measures. In particular, the mixing measure p in the Poisson integral (3.5) arises as the weak limit of Mn. In a special case, coherent families were introduced by Kingman [129] under the name of partition structures. Each central measure is a Markov measure; it is determined by transition probabilities p ( X , A), X / A, defined on the edges of the graph. 1.5. Ergodic method. A harmonic function cp E Harm(r) determines an extreme point of this simplex if and only if the corresponding central measure M is ergodic with respect t o the tail equivalence relation J in the path space T of the graph I?. By definition, the equivalence classes of J consist of eventually coinciding paths. The ergodicity of M means that, for each measurable set A C T consisting of entire equivalence classes of J, its measure M(A) is either 0 or 1. In general there is no natural transformation S : T 4 T whose orbit partition would coincide with the tail equivalence relation I. However, one can restate Birkhoff's pointwise ergodic theorem so as to make the choice of such a transformat ion unnecessary. To this end, in the ordinary statement of the theorem, replace the Cesaro mean f ( S k t ) of a function f : T 4IR by fn(t) =

where the sum is over all paths s E T that coincide with t starting from the nth level r n . A version of the standard argument (see, e.g., [46], Addendum 3) leads to the following important result.

THEOREM 3 [13]. Let M be an ergodic central measure in the path space T of , . . , un, . . . ) such that a branching graph I?. Then the set of paths t = ( u ~u2,.

$1. GENERAL THEORY O F LOCALLY SEMISIMPLE ALGEBRAS

11

(a) there exist the limits

(b) the function cpt: I? is of full M-measure.

~(L,v) lim n+md(O, vn) '

r;

cpt(X)

=

+ E%

is harmonic and related to the measure M by (3.4)

The ergodic method for constructing the boundary of a graph is based on this theorem.

For each extreme harmonic function cp E l(r)there exists a path COROLLARY. t E T such that cp coincides with the function cpt defined by (5.2). It is worthwhile to restate this result in terms of characters.

Ur=l +

COROLLARY [13]. Let E exChar(G,) be an extreme character of a locally finite group G, = G,. Then for each n = 1 , 2 , .. . there exists a character $, of an irreducible representation of the group Gn such that

Let us call a path t E T regular if the limits (5.2) exist. The corresponding limiting function cpt is harmonic, though not necessarily extreme. According to the above corollaries, the computation of extreme harmonic functions and characters can be derived from the description of regular paths of the branching graph.

1.6. Multiplicative branching graphs. Many interesting examples of branching graphs can be obtained by using commutative graded rings; we call these graphs multiplicative. For instance, consider the graded ring R = Rn of symmetric polynomials in infinitely many variables. The Schur functions sx(x), X E yn, form a Z-basis in the additive subgroup R, of polynomials of degree n. By Pieri's well-known formula,

@r==o

and we arrive at another description of the Young graph y . If {fx)xEy is an arbitrary graded basis in R such that the coefficients of all expansions

are nonnegative, we can define a branching graph with vertex set y by declaring that a pair of vertices (A, A) is connected by an edge of multiplicity x(X, A) provided that x(X,A) # 0. We emphasize that t o state the problem of computing harmonic functions on a graph, there is no need to require that the multiplicities of edges be integers. In Chapter 2 we study the multiplicative branching graphs which form a twoparameter deformation of the Young graph y . In this case the basis { f x ) consists of the Hall-Littlewood-Macdonald symmetric polynomials Px(x; q , t ) introduced in [139, 1401.

12

0.

INTRODUCTION

General multiplicative branching graphs satisfy the following extremality criterion. THEOREM4 [126]. Given a harmonic function cp E Harm(r) on a multiplicative branching graph I?, let $: R + R be its extension (by linearity) to the ring R (i.e., $(fx) = p(X) for X E r). Then the following conditions are equivalent: (a) cp E &(I?)is an extreme point of the simplex Harm(r); (b) $: R + IR is a ring homomorphism. Special cases of this criterion were used by Thoma [I611 and Voiculescu [165]. COROLLARY. The boundary &(I?) of any multiplicative branching graph closed in the simplex Harm(r).

r

is

Since all branching graphs considered in this book are multiplicative, the boundary &(I?)is well-defined as a compact topological space, and the harmonic functions are in a one-to-one correspondence with Bore1 probability measures on the boundary.

1.7. The crossed product construction. In Section 1.1 above we have associated a branching graph with an arbitrary family

of finite-dimensional semisimple algebras over @. Conversely, assume that a graded I?, with directed edges enjoys the following properties: graph = (a) each edge (A, A) beginning at the nth level X E r, leads to the next level, i.e., A E (b) 0 E rois the only vertex with no ingoing edges; (c) each vertex has a t least one outgoing edge; and (d) all levels r,, n = 1 , 2 , . . . , are finite. Then it is not difficult to construct an inductive family (7.1) with branching graph r. By definition, the algebra U, in this family consists of matrices of the form

Ur=l

(7.2)

a

=

(a,,),

u , v E Tn(X),

E

r,,

where T,(X) stands for the number of paths of length n ending at X E r,. Define the loop space B = B ( r ) of the branching graph r as the graph of the tail equivalence relation 5 in the path space T. Thus B consists of pairs of eventually coinciding paths s, t E T; a sequence (s,, t,) E B converges to (s, t) if lims, = s, limt, = t in T, and there exists a level rN such that s, and t, coincide for all n N. The loop space B is locally compact and totally disconnected. The limiting LS-algebra U, = 142, can be identified with the algebra of compactly supported locally constant functions a: B + @ on the loop space. The multiplication is defined by

>

where r runs over all paths that are 5-equivalent to s , t. There is a natural involution a*(s,t)

= a(t, s).

This construction of an LS-algebra is a version of von Neuinann's construction of the crossed product associated with a dynamical system (cf. [150]). In particular,

51. GENERAL THEORY O F LOCALLY SEMISIMPLE ALGEBRAS

13

it plays an important role in constructing realizations of factor representations of LS-algebras. For the infinite symmetric group, such realizations were found in [14, 151. Note also that diagonal "matrices" a E % , form a maximal commutative subalgebra of this algebra, and that all characters are states of measure type with respect to this subalgebra:

where p is a central measure on the path space T. 1.8. Dimension groups. A branching graph is associated with an inductive family of algebras (7.1) and depends not only on the limiting LS-algebra a ,, but also on the choice of the approximating family. Though LS-algebras are quite varied, and the problem of their classification is not very interesting, the limiting algebra % , has a nontrivial invariant which reduces all problems concerning LS-algebras to problems related to abelian groups. This invariant was suggested by Elliott [88]; it is the Grothendieck group KO(%,) endowed with a natural ordering structure. By definition, the genera, and tors of this group are the classes [p] of unitarily equivalent projections p E ,% the relations are of the form

where p,q are orthogonal projections. Note that this description uses a specific property of the K-functor on the class of LS-algebras: with the standard definition, the group KO(%,) is generated already by the classes of submodules of the free module with one generator. The semigroup K$(%,) generated by the classes [p] is a cone and defines an ordering structure in the group KO(%,). The class of the unity projection [l]E K$(%,) determines an order unity in the cone K$(%,), i.e., it is contained in no proper order ideal of the Grothendieck group. Elliott obtained the following important result. THEOREM5 [88]. The triple (KO(%,), K$(u,), [I]), which is an ordered abelian group with a distinguished order unity, is a complete invariant of the LSalgebra .Q , Elliott's invariant is continuous with respect t o inductive limits: if % ,

l&

=

an,then

Consequently, it can be easily described in terms of the branching graph I? associated with the algebra 8,. A dimension function on r is any Z-valued function which is defined for almost all vertices of the graph and satisfies the equation

on its domain. We identify two functions f l , fi if they differ only on a finite set of vertices. A dimension function is called virtually positive if it is nonnegative for

14

0.

INTRODUCTION

almost all vertices of I?. A distinguished example of a virtually positive function is the function d(X), X 6 I?, defined in Section 1.4 above. It is easy to see that the group KO(%,) is identified with the group of (classes of) dimension functions; the cone K:(%,), with the cone of virtually positive elements; and the order unity [I], with the function d: r -+N*) . It was a remarkable discovery that Elliott's invariant can be characterized in intrinsic terms. Recall (see [22, 941) that an ordered abelian group G is called a Riesz group if it satisfies the following interpolation axiom: (I) If four elements of the group G are related by the inequalities

fi

>

where y 0, a, 0, pi 2 0, and x(ai+ P i ) a nonzero radius of convergence.

< m. I n particular, these series have

The most difficult problem here is t o prove that H ( z ) = eYZ provided that a totally positive series H ( z ) defines an integral function without zeros. This fact can be derived from the following result of Nevanlinna's. Given a n integral function f , denote by zia),z p ) , . . . the sequence of all points such that f (z!")) = a . By definition, the index of convergence of the value a is the number


0 for all i 1; or (3) pi > 0 for all i 1, implies P ( a ; P) > 0, and (b) if P & ( a ; p) 0 for all a /" v, but P[,l(a; /3) $ 0, then P[,l(a; P) > 0 for all ( a ; R) E Ak,l.

>

-

>

+

THEOREM 15 ([128], 58.1). A vector of real numbers fm(X), X E y m , is virtually positive if and only if the jet of the polynomial (6.3) is positive. 2.7. Branching of Macdonald polynomials. In Section 1.6 we regarded the Young graph as the branching graph of the Schur functions. By using other bases in the ring of symmetric polynomials R we can obtain new interesting branching graphs, including the - graphs which describe the theory of characters of the spin-symmetric group 6, (the group which linearizes the projective characters of the symmetric group 6,) and the branching of conjugacy classes of the symmetric groups 6,.

52. CHARACTERS OF 6,,

AND T H E YOUNG GRAPH

25

In the second chapter we will show that the multivariate Selberg's integrals can be computed as the Poisson integrals representing special harmonic functions on the branching graphs of Jack symmetric polynomials. An interesting class of symmetric polynomials was introduced recently by Macdonald [139, 1401 as a generalization of the more classical Hall-Littlewood polynomials. The Macdonald polynomials Px(x; q, t ) depend on two complex parameters q, t and are determined by the following properties: (a) the expansion of PAin the basis of monomial symmetric functions m x ( x ) ,

is determined by a matrix uxi, which is triangular with respect to the dominance ordering*) of Young diagrams A, p E y,; and (b) the polynomials PAare orthogonal, (7.2)

(PA,

P,),,t = 0

for

# p,

with respect to the scalar product that is defined on the basis consisting of Newton power sums by**)

form a linear For most values of the parameters q, t the polynomials {PA)XEy basis in the algebra R. The branching rule for these polynomials follows from Macdonald's results:

where the multiplicities z q t t ( X , A) are rational functions in the parameters,

Here the product is over all squares b of the diagram X that lie above the "new" square distinguishing the diagrams X and A in the same column. In our notation a = a(b) stands for the arm length of the square b, i.e., the number of squares in X that lie to the right of b in the same row. Similarly, 1 = l(b) is the leg length of b, i.e., the number of squares that lie below b in the same column. Figure 5 shows several first levels of the branching graph y ( q , t) of Macdonald polynomials; it differs from the ordinary Young graph only by the multiplicities of edges. The true Young graph y is obtained as the special case when q = t. If the parameters q , t are real and -1 < q, t < 1, then all multiplicities N ~ , ~ ( XA), , X 7A, are positive, so it makes sense to pose the problem of describing the boundary of the graph y ( q , t ) , i.e., of computing nonnegative harmonic functions on Y(q, t). *) See [49], 51, for the definition. **)We use the standard notation zx = 1'1 r ~2 T!2 r2!. . . , where r, = r,(X) is the number of rows of length i in a Young diagram X E y.

0.

INTRODUCTION

FIGURE 5. Branching of Macdonald polynomials. The extremality criterion from Section 1.6 reduces this problem to the following one.

The generalized problem of total positivity. Find all homomorphisms + R of the algebra of symmetric polynomials R that are nonnegative on Macdonald polynomials:

4: R

(7.6)

$ ( P A )2 0 for all

X E y.

If q = t , the polynomials PA reduce to the Schur functions, Px(x; q, t ) = sx(x), and the problem is completely solved by Thoma's theorem from Section 3.1. In Chapter 2 we will give arguments in favour of the following possible solution of the generalized problem of total positivity.

The formal generating series C O N J E C T U R[119]. E

52. CHARACTERS O F G,,

for homom~orphis~ns 4:R form

+

AND T H E YOUNG GRAPH

27

E% which are positizle in the sense of (7.6) are of the

where

This conjecture is completely proved in the following three cases: 1) for q = t (Schoenberg -Edrei theorem [84]); 2) for q = 0, t = 1 (Kingmall [129, 1301); 3) for q = 0, t = -1 (Nazarov [51]). The latter two examples are related to the classical Hall-Littlewood polynomials ([49],Chapter 111))which are the special cases of Macdonald polynomials when q = 0. In this case the multiplicity of an edge (A, A) is equal to

where r = r(A) is the number of rows of length j in the diagram A, arid j is the length of the row that conta.ins the square b = A \ A. 2.8. The characters and dimension group of a Kingman graph. When q = 0, t = 1 the Macdonald polynolnials reduce to the monomial sylnlnetric functions, (8.1)

Px(z;0 , l ) = nzx(x),

X

E

Y

The coefficients in Pieri's formula

for monomial symmetric functions are positive integers, equal to (8.3)

x(X,A) = rj (A):

where the multiplicities r, (A) are defined in the footnote to Section 2.7. Figure 6 shows several first levels of the corresponding branching graph. In this case the coherence condition for a family of probability distributions M, on the levels y, takes an especially natural form: if we begin with a random diagram A E y, distributed according to Mn(A), pick a uniformly distributed square of this diagram, and remove the rightmost square of the row containing this square, then the resulting diagram X E ynPlis distributed according to hl,-l. Kingman [129, 1301 found the boundary of this graph. His work was motivated by the analysis of the Ewens distribution on the symmetric group G,, which is very popular in populational genetics. We call K: = y ( 0 , 1) the Kingman graph. Handelman [log] arrived at the Kingman graph in a completely different way. He observed that if diagrams X and A have at most m rows, then the coefficient x(X,A) coincides with the multiplicity of the irreducible representation . i r ~of the group SU(7n) in the decomposition of the tensor product of .irx and the natural representation .ir(l) of this group.

0.

INTRODUCTION

FIGURE 6. The Kingman graph. From the viewpoint of the symmetric groups B,, the graph K describes the branching of induced representations of this group. The author's paper [34]contains a description of two equivalent graphs: one of them reflects the branching of conjugacy classes of symmetric groups, and the other describes the branching of partitions of a finite set; they are considered in more detail in Chapter 2, where we also give an approximative proof of Kingman's theorem on the boundary of the graph K ,and compute, for the first time, the semifinite traces and dimension group of this graph. Let us state our main results.

THEOREM 16 (on the Poisson integral). Denote by A the simplex of nonincreasing sequences

with y = 1 - C ai > 0. Define an embedding i: y + A by setting i(X) = ( X l / n , Xz/n,. . . ) for X E Y,. Let the Poisson kernel @(A;a ) be equal to the extended monomial symmetric function m x ( a ; 0; y ) given by

52. CHARACTERS O F G,,

AND T H E YOUNG GRAPH

29

T h e n the implications of the theorem of Section 1.3 hold; in particular, each harmonic function cp E Harm(K) has a unique integral representation of the form

where p is a probability measure o n the simplex A. THEOREM 17 (on semifinite characters). Extreme semifinite nonnegative harmonic functions o n the K i n g m a n graph are of the form

where H , is the differential operator, i n the ring R of symmetric functions, which is conjugate t o the multiplication by the complete homogeneous symmetric function h,, u i s a nonempty Young diagram, and the number of nonzero frequencies in the sequence Q = ( a l , C Y ~ ., . . ) E A i s finite. Let us define the jet of a symmetric polynomial P

E

R as the family of functions

a c A if the conditions for all ( a ; 0; y) E a and a /'

We say that the jet P is positive on a subset cpu(a;0; y) z 0

u

and imply cpu(a;0; y)

> 0 on

a.

$&(a;0; 7 )

#0

on

a

THEOREM18 (on the dimension group). Let P be a symmetric polynomial. T h e following conditions are equivalent: (a) There exists a number n such that all coeficients c~ in the expansion

are nonnegative. (b) The jet of the polynomial P i s positive o n the following subsets of the boundary A for all k = 1 , 2 , . . . :

2.9. Generalized Macdonald polynomials and orthogonal polynomials. A considerable part of Chapter 2 is devoted to studying the generalized Macdonald polynomials. The definition of these polynomials reproduces the definition of the ordinary Macdonald polynomials from Section 2.7 except for the following two points: (1) In formula (7.1) of Section 2.7 the matrix u x P is triangular not with respect to the dominance ordering, but with respect t o a certain total ordering of the set of Young diagrams, which is comparable with the dominance ordering. (2) (7.3) is replaced by a more general bilinear form

30

0.

INTRODUCTION

where w = (wl, w2,. . . ) is an arbitrary sequence of complex numbers, and . ... w,l, = wxl The generalized Macdonald polynomials will be denoted by PA(x;w). Our results may be divided into three parts. The theorems of the first group show that a number of properties of Macdonald polynomials, which Macdonald proved by special techniques, also hold, appropriately stated, for much more general polynomials PA(x; w). An example is provided by the following duality theorem.

THEOREM 19 ([119], Theorem 1). Denote by Qx the symmetric polynomials such that (Qx, P,), = SA,, and let P A be the generalized Macdonald polynomials associated with the conjugate ordering of Young diagrams,

Define a n automorphism w,

of the ring of symmetric functions R by setting

Then

where l / w = ( l / w 1 , 1 / ~ 2 ,... ) . The second group of results characterizes the original Macdonald polynomials in the class of the generalized ones. For example, Macdonald established the following property of his polynomials (i.e., corresponding to w, = (1 - q7') / (1 - t n ) ) : (SO) If X = ( X I , . . . ,An) is a Young diagram with n nonzero rows and :r = (21,. . . , x,, 0 , . . .), then (9.5)

PA(x; w) = 21 2 2 . . . 2, PA* (2;w),

where A, = (A1 - 1 , . . . , A n - 1). We show that this property does not hold for more general sequences w.

T H E O R E20 M ([119], Theorem 2). If property (SO) h,olds for n defining sequence is of the form

=

2, then the

where c is a constant, and the sequence {w,} is given by one of the following formulas:

(9.8) (9.9)

wn=(Y, wkm = a ,

TLXl,2, . . . , W, = 1 if n $ 0 mod rn.

Note that (9.8) and (9.9) are limit cases of (9.7). The results of the third kind establish a relation of Macdonald polynomials Px(x; q, t ) to the family of Rogers-Ramanujan orthogonal polynomials [154].The latter are obtained as follows. Substitute two variables x = ( x l , 2 2 ) into Px(x; q, t ) , then set 21x2 = 1, and consider PAas a polynomial in one variable y = xl r 2 .

+

53. T H E PLANCHEREL MEASURE O F

6,

31

This observation allows us to relate the characterization of the true Macdonald polynomials among the generalized ones to an old problem of Fejkr's [go]. This problem is stated as follows: among all polynomials of the form

where bo, bl, . . . is a scalar sequence, find orthogonal polynomials with respect to a certain measure. Fejkr's problem was essentially solved in [60] and [48]. We establish the equivalence of these two problems, and correct some inaccuracies contained in [60] and [48] (the case (9.9) in o m notation was overlooked there). The solution is based on the following unexpected fact.

THEOREM 2 1 [119]. The general solution of the infinite system of nonlinear equations

in variables f l , f 2 , f 3 , . . . depends only on three arbitrary parameters: if f 2 and f3 # f l , then fn = f l ( f 2 - f l ) ~ , , where

+

and Un(cos ip) = sin(n kind.

+ 1)ip / sin ip are the

# fl

Chebyshev polynomials of the second

In conclusion, we would like to draw the reader's attention to Theorem 6.3 from [119], which provides a new determinantal representation of certain Macdonald polynomials Px(x; q, t). These formulas were apparently not known even in the case of mono~nialsymmetric functions, i.e., q = 0 and t = 1. 53. The Plancherel measure of 6,

In Chapter 3 we will study the most interesting character of the group 6,, the character of its regular representation. This is simply the &function supported by the unity; however, its dual description in terms of the branching graph leads to the very important Plancherel growth process of Young diagrams. Our main result in the third chapter is a central limit theorem for the Plancherel measure. 3.1. Definition of the Plancherel measure. The Plancherel measure of a finite group G is*) a probability distribution on its dual object**) whose weights are proportional to the squared dimensions of irreducible representations. It is clear from Burnside's formula

*)See, e.g., [43], Section 12.4. **)I.e., on the finite set of equivalence classes of irreducible unitary representations.

32

0.

INTRODUCTION

that the weights of the Plancherel measure are equal to

Plancherel's theorem for the ordinary Fourier transform

claims that

(where tr(a) = C aii is the normalized trace of a matrix a E Mn@),and the Fourier inverse can be written in the form

A

If f , g are central functions, then f (n), ?(n) are scalar matrices, and it is convenient to regard them as scalar functions on G. Formulas (1.3), (1.4), (1.5) take the form A

A

where xT(w)/dimn is the normalized character of n E G. In particular, for w = e we obtain the identity

For an infinite discrete group G, there is also a standard definition of the However, Plancherel measure as a probability distribution on the dual space for wild groups similar to 6, this definition is not interesting. We suggest a new definition of the Plancherel measure for the class of locally finite groups. Fix an approximation G, = 1 4 G n of a locally finite group G, by finite subgroups G,, and let r be the corresponding branching graph.

e.

DEFINITION. The Plancherel measure of the group G, (with respect to the chosen approximation) is determined by the Markov chain on the branching graph r with transition probabilities JG,I p(A7A)= -. IGn+lI

x(A,A)dimn~ dim nx

'

53. T H E PLANCHEREL MEASURE OF 6,

where X E I', A E I',+l, and X / A. The initial state is the unique vertex 0 This Markov chain will be called the Plancherel growth process.

33

E

Fo.

This Markov chain is well-defined, as follows from counting the dimension of the induced representation

in two ways. We obtain

Gn+ll

lGnl

dim aA=

C

x(~ A), dim A,

A: A/A

whence CAp(X, A) = 1. Denote by Mn the distribution of the state X E r, of the Plancherel growth process after n steps. It is easy to check that Mn coincides with the Plancherel measure for G,,

which explains the choice of the term. 3.2. The typical shape of large random Young diagrams. The Planchere1 measure of the infinite symmetric group 6, is determined by the Markov chain on the Young graph with transition probabilities

This is an ergodic central measure on the space of infinite Young tableaux T; it corresponds to the zero frequencies a1 = a 2 = . . . = Dl = p2 = . . . = 0. Important results on the asymptotics of the shape of Young diagrams in the course of the Plancherel growth process were obtained by Vershik and the author [12], [18],and independently by Logan and Shepp [138]. Most problems considered in this book are motivated by these results. It will be more convenient to discuss the limit shape of Young diagrams if we slightly modify their original description. Following the combinatorial tradition, we have been believing up t o now that Young diagrams are merely graphic representations of partitions of positive integers*). Now we will change our viewpoint and regard Young diagrams as piecewise linear functions (Figure 7).

FIGURE7. Two ways t o draw a Young diagram *)Another generally accepted way of representing partitions, Ferrers diagrams (see [49], p. 17), is completely useless for our purposes.

34

INTRODUCTION

0.

DEFINITION.A Young diagram is any continuous piecewise linear function v = w(u) that has the following three properties: (1) wl(u) = f1; (2) w(u) = lul for sufficiently large lul; (3) the points of minima X I , .. . , xd and the points of maxima yl, . . . ,yd-1 of w are integers. For example, a one-square Young diagram w E yl corresponds to the function w(u) = max(lu1, 2 - l u l ) In general, the number of squares of a diagram w E yn can be written in the form

If we want to obtain a nontrivial limiting function for growing Young diagrams, we should uniformly rescale them t o keep the square of the subgraph (2.2) equal to one. Consider the mean value of rescaled Young diagrams with n squares with respect to the Plancherel measure Mn:

THEOREM 22. The uniform limit R(u) = lim wn(u) of the means (2.3) exists, n-03 and it coincides with the function

Since R1(u) = $ arcsin ,; we will call R(u) (as well as every statement which claims the uniform convergence to this function) the arcsine law. The asymptotic behaviour of the Plancherel means can be deduced from the following strong law of large numbers. THEOREM23 [12]. The uniform limit 1 lim --wn(u&)

n-w

+

= R(u),

where R is the arcsine law (2.4)) exists for almost all, with respect to the Plancherel measure, infinite Young tableaux t = (wl, wz, . . . , w,, . . . ) E T . Figure 8 shows the graph of the function R along with a random 100-square Young diagram obtained by computer simulation of the Plancherel measure. A remarkable consequence of the theorems on the limit shape of Young diagrams is the solution of Ulam's problem [I631 concerning the length Ln(x) of the longest increasing subsequence x = {xj),R=,. COROLLARY [12, 181. Let x be a sequence of 2.i.d. random variables with a common continuous distribution. Then, for every E > 0, (2.6)

{

lim Prob x:

n-00

IL$'

-21

-

0 and k = 1 , 2 , .. . , set

T h e n CApY)(X,A) = 1 for all X E y , and the probabilities (1.7) determine a central Marlcov chain o n the k-row part Y(k) of the Young graph. Thus we have justified the derivation of Selberg's integral (6.10) in Section 3.6. The more general integrals (6.14) in $3 can be obtained in a similar way; in this case the transition probabilities are also given by (1.7)) but instead of the parameters xk, y k of the original Young diagram one should use the parameters xk(0),yk(0) of the rectangle diagram X(0) obtained from X by rescaling by 0 along one of the axes. This is one more argument in favour of taking a serious attitude toward nonintegral rectangular diagrams.

4.2. Asymptotic behaviour of interlacing sequences. How can a pair of interlacing sequences

behave as n grows? In order to make the question more precise, consider the rectangular diagram w, with minima points {xk) and maxima points {yk); it is easy to see that this diagram is uniquely determined. We can now restate the

46

0.

INTRODUCTION

problem as follows: what is the lirnit shape of the diagrarn w,? We saw in $3 that the lirnit shape of typical Young diagrams with respect to the Plancherel measure is described by the function

+d

arcsin

m )

if

lu1 5 2,

if

lu1

> 2.

Here we will show that the sarne "arcsine law" holds in purely analytical situations which seem cornpletely unrelated to the symmetric group. In our first example we consider a family of orthogonal polynomials {Pn(u)}r=o defined by a linear recurrence relation

and initial conditions Po(u) = 1, PI(u) = u the adjacent polynornials

-

bl. It is well known that the roots of

interlace. To describe the character of mtittial separation of the sequences {xk), {yk), we use the rectangular diagram w, defined by the following conditions: (2.5)

P n 1(u) wk ( u ) = sign , wn(u) = Iu P n (u)

-

Cn 1

for sufficiently large lul,

where Cn = C - C y,. It turns out that for a wide class of orthogonal polynomials (including the classical polynomials of Jacobi, Laguerre, arid Herrnite) the asymptotic behaviour of interlacing roots obeys the arcsine law. THEOREM 33 [38]. Assum,e th,at th,e coeficients of th,e recurrence relation (2.3) satisfy th,e conditions

Then th,e linrit 1 lirn -w,(~c,-~ cn

n-oo

+ b,)

=f~(u)

exists uniform,ly in u E EX, th,e function fZ being given by (2.2) Figure 11 shows the diagrams of root separation for the Chebyshev, Hermite, and Laguerre polyriornials of degrees 15 and 16. The graph of the function (2.2) (appropriately rescaled) is laid upon the diagrams. The asymptotics of the separation of roots should be distinguished from the asyrnptotics of their distribution. For example, the lirnitirig density of roots of the Jacobi polynomials is described by the inverse semicircle law, 1 lirn -#{i: xi

n-oo

n

< u)

=

dn: -

IuI

I 1;

54. CONTINUOUS YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS

-1.0

-0.5

0.0

0.5

47

1.0

Chebyshev polynomials

Hermite polynomials

Laguerre polynomials FIGURE11. Separation of roots of orthogonal poplynomials.

and the same density for the Hermite polynomials is described by the semicircle law,

Thus the limiting densities of roots are different, but the arcsine law holds in both cases. It is not difficult t o give sufficient conditions for the arcsine law in terms of the measure ,u with respect to which the polynomials P, are orthogonal. For example, the law holds if the measure is absolutely continuous on [-I, 11 and the function log,ul(cos0) is summable on [0, T I . The second example of an appearance of the arcsine law in an analytical problem is related to Wigner's classical semicircle law (see [142], [55]), which claims that in the typical spectra of large random matrices the eigenvalues are distributed with the semicircle density 2d-l~. Consider a stable mechanical system with n degrees of freedom which oscillates near the equilibrium. Its potential energy is quadratic and determined by a symmetric positive definite matrix A. The squares of the fundamental frequencies of the system coincide with the eigenvalues X I ,. . . , x, of this matrix.

0.

48

INTRODUCTION

If a linear coristrairlt is imposed on the systern, the11 the new fundamental frequencies interlace with the initial ones, so that the separation conditiorl (2.1) holds. This result is know11 as Rayleigh7s theorern (see [I],$24). The rectangular diagram w, that describes the interlacing spectra of an Hermitian matrix A and its restriction to a certain hyperplane h will be called tlle rigidity diagram. THEOREM34 [38]. Let w, be the rigidity diagram of a random symmetric matrix A(,) of order n, with respect to a random hyperplane 11. c I%,. Assume that the normal to h is uniformly distributed on the unit sphere ofRn, and the matrix ( entries ai:), i j , are independen,t of 11. an,d among themselves, and identilnlly


m.

,

The degree of P,

as a polynomial

The proof of this lemma may be found in the book [50](Chapter 6, 51, Section 5); it is not difficult to compute the polynomials Pmexplicitly. The lemma also holds for arbitrary permutations a E 6,; for example, for a = (1,2) (3,4), X" (am) dim X

-

~PT

P; (A) - 4 ~(A) 3 + (A) - 3Pl (A) n ( n - 1) (n - 2) (n - 3)

Relations of this kind will be considered separately. *)1n Figure 3, p. 21, r(X) = 5,

f = ( l l ; , 9;, 7;,

3;, 2;);

g = (7+, 4:,

3;, I;,

;).

52. CHARACTERS O F Bm

COROLLARY 2 . The following asymptotic formula holds:

where the error term tends to zero as n 4 ca uniformly i n X E Yn. The proof follows from the homogeneity of p, and the estimates

LEMMA2. The following conditions o n a tableau t E T are equivalent: 1 ) For each m > 2 there exists the limit lim

n-m

xtn

"(a,)

dimt,

- Pm

2 ) For each k = 1 , 2 , .. . there exist the limits fk(tn) lim =ak,

n-m

n

lim gk(tn) n = Pk.

n-oo

If these conditions are satisfied, then

PROOF. If the limits (2.2.8) exist, then by Corollary 2

pm = lim

n+m

Conversely, the sequences recovered from the sums p,:

xtn " m ) dimt,

> a2

a1

=pm(al,..

. :PI).

2 . . . and pl 2 jj2

> .. .

can be uniquely

if

and

then

Hence if for some k the sequence fk ( t n ) / nhas more than one limit point, then the limits (2.2.7) do not exist either. Thus the numbers a k and Pk have a clear meaning: they are the frequencies of the squares of the kth row and kth column, respectively, in the growing tableau. As follows from the main theorem, the character is determined only by these frequencies.

78

2. BOUNDARY O F THE YOUNG GRAPH AND MACDONALD POLYNOMIALS

2.3. Parametrization of characters.

THEOREM 2 (Thoma [161]). All normalized (indecomposable) characters of the group B, are given by the formula

where a1 > a 2 . . . L 0, PI 2 P 2 > . . . > 0, Cai+CPi < 1, and p,(a) is the number of cycles of length m i n a permutation a . I n particular, if a, is a permutation with a single nontrivial cycle of length m, then cp,,p(am) = p,(a,P). Thoma proved this theorem by using, first, a simple fact on the multiplicativity of indecomposable characters of B, with respect to cycles, which reduces the problem to computing the values of the character on one-cycle permutations; and second, the fact that the generating function for these values satisfies a certain functional equation. Thoma solved this equation using a deep theorem of Nevanlinna's [144] from the theory of integral functions. The meaning of the parameters a, which index the solutions of this equation has no explanation within this argument. Our proof is based on the ergodic method and passing to the limit from finite groups.

PROOFO F THE T H E O R E M . Let cp be a normalized character. By Theorem 1 there exists a tableau t E T such that ~ ( o = ) lim for all o E 6,. By Lemma 2 the limits (2.2.8) exist, and cp(a,) = pm(al, . . . ; PI, . . . ). Finally, according to the above-mentioned multiplicativity, the value of the character on an arbitrary element a E B, can be expressed as the product of its values on one-cycle permutations: cp(a) = - cp(a,)~m(~).

nm,2

Note that the generalization of (2.2.5) to arbitrary permutations yields (2.2.9) directly for an arbitrary element a E B,, without using the multiplicativity. Thoma's multiplicativity theorem, as well as its generalization to the group U ( m ) due to Voiculescu, is a consequence of the following general theorem: A n R-valued indecomposable positive functional on a Riesz ring is a ring homomorphism. The dimension group of the algebra @[B,] is a Riesz ring (see [126]). Denote by M,,p the central ergodic measure associated with the character cp..,p by formula (6). p, ( a , P ) P m , where p is a Young diagram with The functions pp(a,P) = ) given by (2.2.4), are called the extended row lengths pl,p2,. . . , and p m ( a T ~ is Newton functions. Since all symmetric functions can be expressed in terms of Newton functions by well-known formulas, they give rise to the corresponding extended functions in a, p. In particular, the extended Schur functions s x ( a ,P) are defined by Frobenius' formula

n,,

pP(a,B ) =

xA(p)

0).

A

Comparing this formula with (6) yields

COROLLARY 3. The values of the central ergodic measure Ma,8 on the cylinders F,, u E T(X), are given by the extended Schur functions: M..,~(X)= s x ( a ,P).

52. CHARACTERS OF 6,

lim

79

COROLLARY 4. For almost all, with respect to Ma,p, tableaux t E T the limits = a k and lim = pk exist for all k = 1 , 2 , . . . .

The most surprising consequence of Theorem 2 is that the zero frequencies a = p = 0 correspond to the unique character S,, the character of the regular representation. The corresponding central measure, called the Plancherel measure, will be studied in detail in Chapters 3 and 4. It turns out that the rows and columns of almost all tableaux with respect to this measure grow as fi,and after an appropriate scaling (1/fi) the shape of the tableau becomes stable. Thus, to obtain all limiting characters, it suffices to consider only tableaux with row and column lengths either linear in n, or of the specific order fi. The parameters a, ,D have another interpretation related t o the RobinsonShensted-Knuth (RSK) correspondence and the construction of the representation of type 111 associated with the character p,,p (see [14]). In this connection, we mention here only the following fact. If p is a Bernoulli measure in the space of sequences with a common distribution of coordinates a = (al,a 2 , . . . ), CEl ai = 1, then the value p a I o ( a )is the p-measure of the set of fixed points with respect to the permutation a of coordinates:

This interpretation is related to another proof of Thoma's theorem suggested by Vershik and the author in [127]. Namely, there exists a map from the space of IR into the space of tableaux T (the generalized RSK-transform) sequences X = such that each central ergodic probability measure on T is the image of a product measure. LetX~Y~beaYoungdiagram a n, d f i = fi(X)+;,iji = g i ( ~ ) + ; , i = 1, . . . , r . Set 6f = Bf1 x Bj2 x . . . , 6 9 = Bg1 x Bg, x . . . , and let 4' be the character of en+,induced from the Young subgroup 6f x 6 9 by the character 1x sign, where 1is the identity character, and sign is the signature character. It is not difficult t o check that if the frequencies (2.2.8) exist for a sequence of diagrams {t,), then

ny

Gtn (a) lim Gtn ( E ) = cpa,p(u)

n+,

for all a E 6, (here E is the identity permutation). Thus Theorem 2 implies COROLLARY 5. Allfinite characters of 6, are the limits of induced characters. The direct (independent of Theorem 2) proof of this corollary uses the abovementioned generalized RSK-correspondence; it has been published in [127]. It follows from (2.2.11) that for a large diagram X the dominant part (in the sense of relative dimension) of the decomposition of the induced representation with character 4x consists of irreducible representations with characters close to X A . This statement may be called the law of large numbers for induced representations. It seems to be of a quite general character. 2.4. Semifinite characters of 6,. In this section we compute all infinite characters of the group 6,. This allows us to describe the cone of positive elements in the KO-functor of this group.

2. BOUNDARY O F T H E YOUNG GRAPH AND MACDONALD POLYNOMIALS

80

Given a tableau t E T(Iky,'), denote by vt : dk>') -+ W the restriction of Nt to v("') = v (k, I)*). Let f E T(Ik,') be a maximal strictly increasing subsequence in the sequence of diagrams {t, n Ik,L)p=l. The tableau t can be recovered from f and vt, so that the whole space of tableaux on Iky,' is partitioned into the union U T, of disjoint parts T, = {t : vt = v), where v : u ( ~ > '+ ) W is a monotone embedding, each T, being embedded in T(Ik,'). Given a finite collection of numbers a1 . . . a k > 0, 2 ... > 0, ~Zk_,a(CfzlPi= 1, let Ma,p be the corresponding central ergodic probability measure on T(Ik,'). Define a measure M:,p on T(IL 1) as follows: on each T, it is the inverse image of M,,B under the embedding t + f. In other words, if u is a Young tableau of shape X and v(k)l)c A, then by definition

+

>

>

+

>

If u, w are Young tableaux of shape X and c A, then u, ZZ E T(XnIk,1)and Ma,p(F,) = M a , ~ ( F a )so , that the measures M:,p are central. Their ergodicity is also obvious, since any Young tableaux u E T(X1), w E T(X2) can be extended t o tableaux with a common diagram X = X1 U X2. It is not difficult to check that the ) [ t] c k Iy l . poles of the measure M,",p are precisely the tableaux t with u ( ~ , ' @ We will show that each infinite central ergodic locally finite measure is pioportional to one of the measures M&. ~

(

~

1

'

)

LEMMA3. Let M be a central ergodie locally finite or finite measure. Denote by IME yoothe union of all diagrams X c N2 with M ( X ) # 0. Then M(A) # 0 for all A c I M .

-

PROOF. If M ( X ~ ) , M(X2) # 0, then M(X1 U X2) # 0; otherwise {t : [ t ] > X I ) and {t : [ t ] 3 X2) are disjoint J-invariant sets of positive measure.

LEMMA4. Let {t,} be a sequence of diagrams such that the limiting frequencies Pi = 1. Then m, = lvnl = o(n) and (2.2.8) exist and a k , P1 > 0, C ; ai

+

(2.2.13) as n

+

dim(& t,)

-

nmn

-dim vn dim p,

m,!

co, where p, = (tn\X) n 4,1, v,

=

(t,\X)

\ Ik,l, A E Y

(Figure

4,

p. 23).

PROOF. AS described a t the beginning of this section, we can associate with each tableau w on t,\X a p,-tableau u and a monotone embedding v : v, 4 { I , .. . , n) which uniquely determine w. Thus dim(X, t,) does not exceed the total number C,"" dim v, dim p, of all such pairs (u, v). Pick a subsequence r, -+ m such that m,/r, 4 0, r,/n 4 0 as n + co. Let T, be the set of p,-tableaux u m,, g1 (urn) m,; it follows from (2.2.8) that lim ITn(/ dim p, = 1. with fk(ur,) Finally, if u E T, and v : v, + {r, 1 , . . . ,n ) is a monotone embedding, then the pair (u, v) corresponds to a certain (t,\X)-tableau w, so that dim(/\,t,) 2 dim v, . IT, 1. Both estimates on dim(X, t,) are equivalent as n + co to the right-hand side of (2.2.13).

>

he sum is taken in Z2.

>

+

52. CHARACTERS OF Bm

81

THEOREM3. Each locally finite ergodic central measure M o n the space of tableaux T is proportional to one of the measures M&. PROOF. Let X be a Young diagram with 0 < M(X) < co. By Corollary 1 there exists a tableau t E T such that (2.2.2) holds. Dropping down to a subsequence, if necessary, we may assume that for the sequence of Young diagrams {t,)p?lthe limiting frequencies a = ( a l , a 2 , . . . ), ,B = (PI,B2, . . . ) in (2.2.8) exist; let m be the corresponding probability measure. Then lim

dim(A, t,) dim t ,

= m(A)

AEY.

for all

If m(X) # 0, then dim(A, t,) dim t, m(h) +dimt, dim(X, t,) rn(X) ' so that M(A) = am(^) and M is a finite measure. Thus the case I, = N2 is not interesting, and we may assume that I, = Ik,'. Suppose that m(X) = 0, i.e., that X\Ik,' # 0. Then v = X-('">')is a nonempty Young diagram*),and by Lemma 5 we have

for every Young diagram A with A-("') = v. Comparing with (2.2.2), we see that the measures M and M& are proportional. Going over from central measures on T to characters of 6, by general formulas, we obtain the main result of this section.

>

>

THEOREM4. Let a1 _> . . . a k > 0, 2 . . . PL> 0, and let v be a nonempty Young diagram. The formula

C: ai +

~i

=

1,

c X c I;,",' determines an indecomposable semifinite character of the group em**). If x-(~%')c v , then pE,,p(EA)= co; for X $ I;,, we have pL,p(Ex) = 0.

for

~

(

~

1

'

)

#

Each (infinite) semifinite character of 6, the characters p i , p .

coincides, up to a factor, with one of

Note that the character p&, v E Ym,can be described as the character induced from the subgroup 6, x 6, of permutations a E 6, with u(k) < m for Ic < m:

*)M-(~= - ~N2 ) ( p - ( k , l ) ) , where the subtraction is in 2 '. * * ) ~ e c a lthat l sA(a, P ) is the extended Schur function.

82

2. BOUNDARY OF THE YOUNG GRAPH AND MACDONALD POLYNOMIALS

$3. Generalized Macdonald polynomials and orthogonal polynomials 3.1. The Hall-Littlewood polynomials were originally introduced by P. Hall [I041 and D. E. Littlewood [137]. An excellent exposition of their properties and numerous applications to the theory of (ordinary, projective, modular) representations of symmetric groups and full linear groups can be found in [49]; see also [143]. In a recent paper [139], Macdonald introduced a more general family of symmetric polynomials and described their remarkable properties. As a limiting case this family contains Jack polynomials, in particular, zonal spherical functions of the symmetric spaces GL(n, R)/O(n) (cf. [112]). Combinatorial properties of Jack polynomials were also studied by Stanley [159] and Hanlon [106]. The main purpose of this section is to characterize the HL-functions introduced by Macdonald within a much wider class of polynomials which depends on infinitely many parameters. In a certain sense, the latter are infinite-dimensional versions of the generalized Legendre polynomials defined by Fejkr [go]. From this viewpoint Macdonald HL-functions correspond t o continuous q-ultraspherical polynomials, first considered by Rogers [154] in his famous paper on the Rogers-Ramanujan identities. 3.2. Let us describe our approach in more detail. In this section we borrow the notation and terminology concerning partitions and symmetric functions from 1491. Let h = @ A, = A 8 @ be the graded algebra of symmetric polynomials with complex coefficients in variables x = (xl, x2, . . . ), and let P = U P, be the set of partitions of positive integers. Given X E P, denote by px the corresponding monomial symmetric function, and by px the corresponding Newton power sum. Consider the family of symmetric bilinear forms (., .), on A, depending on a sequence of complex parameters w = (wl, w2,. . . ), which is defined by the formula

where ZX =

l m l m l !2m2m2!.. .

and

WX

=

wX2

for a partition X = (lm12mZ. . . ) with multiplicities m l , ma, . . . . We fix an arbitrary total ordering 5 on P, that agrees with the natural (dominance) ordering, for example, the reverse lexicographic ordering. Define the generalized Hall-Littlewood polynomials (GHL-polynomials, for short) Px(x;w) as the result of the orthogonalization process applied t o the basis {mx(x)) with respect to the form (., .) and the chosen ordering. Thus Px(x; w) = mx(x)

(PA, P,), Denote by Qx(x;w)

= bx(w)Px(x;w)

+

uxP(w)mp(x), P

w h =a;

=

W,

1 if

n

# 0 mod

m.

As far as I know, this case was not earlier considered in this context. For m = 1 we arrive at the ordinary Jack case (J). (M) Macdonald case:

This case was studied in 11391. All previous examples are special or limiting cases of this one. For instance, in order to obtain the case (J,), one should take q = r + a . E, t = r E, where r is the mth primitive root of unity, and let E -, 0.

+

3.4. We present a list of basic properties of the HL-polynomials which were proved by Macdonald in the case (M). By continuity these properties are also valid in the Jack cases (J) and (J,). (i) Duality: w, PA(%;w) = Qxf(x; l / w ) for the algebra endomorphism w, : A + A defined by w,(p,) = (-l),+' . w, . p,, n = 1,2, . . . . Here A' is the conjugate partition of A and l l w = ( 1 1 ~ 11, 1 ~ 2 ,. .. ). (ii) ~ u ~ e r o r t h o ~ o n a l i t yLet * ) : A = (A1,. . . , A,) be a partition of length n and x = ( x l , . . . ,x,,O,O,. . .); then

Px(x; w)

= 21x2.. .x,

where A, = (A1 - 1 , . . . , A, (iii) Supertriangularity:

-

. P A * (x; w),

1).

uxP(w) # 0

implies p l

We will compute the Cauchy kernel

for x = ( x I , x ~. ., .), y = (yl,y2,. . . ) in terms of W(z).

LEMMA.n

( ~Y; ,W) = n,,,w ( x i y j ) .

PROOF.Since

we have

- Y I 2 (fi) Y u l (m) Y g

3-

. s ~ o ~vurura1 ~ o j ayl puv

%)a

(2

a?ON '

(x)Y z u (m) Y

g

3 = (mix)b,

'dOOUd

.vliuvu37

C'z

:l u a p -~!nba axv suo!?!puo3 3u!~o11ojay? ?vy? ysqqslsa ue3 auo ' [ ~ ' P - I' 6 ~ 1u! se Bu!n8.1~ .v vqga31.t. papel3 ay? jo { " ( L ) ' { Y n ) sasvq Jvauq snoaua8ouroy OM?xap!suoD S T V I N O N A T O ~~ T V N O aav ~ ~ H V H~~ X E I~ N ~ O ZH.L A

do

A x v a a n o a 'Z

z6

56. DUALITY

93

56. Duality 6.1. In the previous sections we introduced the GHL-polynomials Px(x;w), Qx(x;w) using the chosen total ordering in the set P,. Consider now the conjugate ordering I' defined in Section 4.3 and denote by P i ( x ; w), Q i (x; w) , bi (w), etc., the corresponding objects. We will use the endomorphism w, from Section 5.3 and denote the sequence ( l l w l , 1 1 ~ 2 ,. . ) by l l w . The main result of this section is




A/)

=aX,;

then the minor (?l;i') differs from in two respects: the rows are arranged in the reverse order, and the entries of the columns corresponding t o odd partitions p E P, have opposite signs. Hence

94

2. BOUNDARY O F T H E Y O U N G G R A P H AND MACDONALD POLYNOMIALS

where E ( R )= (2.6.3) that

npER ~ ( pand ) t(R)

Since w R w R =

nw p

=

C p E R ( N- r ( p ) ) , N

=

IP,(. It follows from

= f,(w), the lemma follows.

6 . 4 . L E M M A .For all A, p E P,,

PROOF.Consider the partition P, = M U { A ) U L of the set P,, where M = {p' : p 2

and real f l , f,

fl)/k

i f n = km, otherwise

101

# 0, fm # f l , the

and (2.7.9), respectively, satisfy equations (2.7.8).

PROOF.Denoting O

= f,/

f l , write the sequence f , in a more explicit form:

It is clear from (2.7.9) that the coefficients cn are given by (2.7.6), and the proposition follows by direct computation. We will call (2.7.15) the complementary series of solutions of (2.7.8). This fanlily was overlooked in [48, 601; in [65] it appears in another form. The complementary series can be obtained as the limit of solutions of the principal series (2.7.13). For this purpose, denote by r a primitive mth root of unity, and set

Substituting these values into (2.7.14), we obtain (2.7.16) in the limit as S -+ 0. For m = 1 the solution (2.7.16) corresponds to the ordinary Jack case (J). It can be formally included into the principal series (2.7.14) by setting y2 = 4.

7.7. Let us show that the principal and complementary series exhaust all solutions of (2.7.8). To begin with, note that equations (2.7.8) and (2.7.9) are homogeneous; hence the set of solutions is invariant under the transformations

a # 0. Since b, = fl f2 . . . f,, we obtain b p ) = anbn, w ( " ) ( z= ) W ( a z ) , and w t ) = w n / a n for n = 1 , 2 , . . . . It is clear from (2.7.9) that the polynomials R, ( y )/b, are invariant under (2.7.18).

THEOREM. Propositions 7.5 and 7.6 exhaust all solutions of (2.7.8), (2.7.9) up to a nonzero common factor.

PROOF. It follows easily from (2.7.8) and (2.7.9) that

for all 1 < k

< n. Assume that for some k the factor

does not vanish. Then fn+l and c, are uniquely determined by the values f l , f 2 , . . . , f,. We will find such factors for all solutions of the complementary series. Assume that f l = f2 = . . . = fmPl = 1 and O = f , # 1. For k = m a n d s 2 we have: 1. if n = ms, then

>

M = 1 - f m / f,,

=

(S -

1) ( 1 - O ) / ( O

+ s - 1 ) # 0;

102

2. BOUNDARY O F T H E Y O U N G G R A P H A N D MACDONALD POLYNOMIALS

2. if n

= ms

-

1, then

h i ' = 1 - fin . fms-, 3. if n = rns - t , 1 < t

=

(1 - 8) (8

+s

-

l)/(s

-

1) # 0;

< m, then M=1-fm=l-QfO.

It remains t o consider the case m = n, rn 2 3, where we set k = 2. Then 4. M = 1 l / fn = (8 1)/8 # 0. We conclude that the initial values f l = . . . = fin-l # f,, m 2 3, uniquely determine the solution { f,}, and that this sollitiori belongs t o the corriplerrleritary series. Consider now the cases f 2 # f l or f3 # f l . If f 3 = f 2 # f i , then for k = 2 and r~ 3 the factor (2.7.20) does not vanish: M = 1 - f 2 / f l # 0; we obtairl the Jack case ( m = 1). The cases f l = f3 # f 2 arid f l = f 2 # f 3 reduce to those considered in (1) and (4), and deterrrli~iesolutions from the corriplementary series with rrL = 2 and m = 3, respectively. 111 the general case f l # f 2 , f l # f 3 , f 2 # f3 the solutio~i{ fn} is uniquely deterrriined by f l , f 2 , f3 (the principal series), see Section 7.5. The proof is complete. -

-

>

7.8. Let us returri t o symmetric GHL-polyriorriials. It follows from (2.6.22) that

for appropriate coefficients cn = c,(w) = WL(,),(,,~)(W).Assurriirig that x = ( x l , x2,0, . . . ) and using the superorthogonality conditiori (ii) frorri Section 3.4, P(n-l)(x; w), and Corollary 7.2 implies tlie folwe obtain that P(n,l)(:c;w) = lowing:

PROPOSITION. The superorthogonality condition (ii) of Section 3.4 in~pliesthat th,e generalized Legendre polynornials'b Rn(y) associated with symmetric GHL-polynornials sati.sfy the recurrence relations (2.7.5).

>

Assuming that c, 0 for all n = 1 , 2 , . . . and using the rerrlarks from Section 7.4, we can assert that tlie polynomials Rn(y) from the proposition are orthogonal as polynomials in one variable. From tlie last proposition and Theorem 7.7 we obtain our second main result.

THEOREM. Assume that GHL-polynornia1.s Px(x;w) satisfy condition (ii) frorr~ Section 3.4 with n = 2. Then the sequence w(') = {cn w,}, for an appropriate constant c, 2.7 of the form (M) or (J,) (.see Section 7.3). 58. Determinantal formulas for GHL-polynomials

8.1. In the rerrlairider of this chapter we consider GHL-polynorriials only in the Macdonald case (M) and use the notation of [139]:

for wn = (1 q n ) / ( l - t n ) , r~ = 1 , 2 , . . . . We would like to obtain deterrrlinantal formulas for PA,Qx that in the Sclmr case (S) wo~ildreduce to the well-known second Weyl formula: -

58. DETERMINANTAL FORMULAS FOR GHL-POLYNOMIALS

103

From another viewpoint, these formulas generalize the classical representations of orthogonal polynomials by determinants of three-diagonal Jacobi matrices:

+

where c, are the coefficients of the recurrence relations y . R,(y) = Rn+l(y) C , R , - ~ ( ~ )determining the polynomials Rn(y). By continuity, our formulas are also valid in the generalized Jack case (J,). 8.2. Let us introduce the rational functions 1 - qAi-i+j tm-j X cij(9, t ) = 1 - q ~ tm-i i '

where X = (A1, . . . , Am) is a Young diagram with m nonzero rows. We will study the determinants (2.8.4)

m

YA(X; 9, t) = det(c,?j(q,t) e x % - i + j ( ~ ) ) i , ~ = l

and

If q = t, then c $ ( ~t) , = 1, and the determinants cpx = expressions (2.8.1) for Schur functions:

turn into the ordinary

since q,(x; q, q) = hn(x). Following [139], denote by w , , ~ the automorphism of the algebra A defined by

then wt,, is the inverse automorphism. By the duality theorem ([139], (5.2)), wt,, en = ~

t , P, ( l n ) (x; t, 9) = Q(n) (x; 9, t) = qn(x; Q,

t)

and wqVtqn(x;q, t) = en(x). Applying the automorphisms w,,t, wt,, t o the determinants cpx, $A yields

8.3. There are natural generalizations of (2.8.6) and (2.8.7) for hook diagrams.

THEOREM.Let X = (n - k, l k ) be a hook diagram with arm length n and leg length Ic. Then (2.8.10) (2.8.11)

Qx(x; 9, t) = $x(x; 9, t),

PA! (x; q, t) = cpx (x; q, t).

-

k

-

1

2. BOUNDARY O F THE YOUNG GRAPH AND MACDONALD POLYNOMIALS

104

PROOF. Macdonald [139] obtained explicit formulas for the functions b, (q, t) in particular, for p = (1")

= b,(w);

Expanding the determinant (2.8.5) by the first row yields

, t )Q ( l j ) ( ~ ; q , tfor ) all j By induction on n, assume that $ ( l j ) ( ~ ; ~ = follows from Theorem (6.5) in [139] that

< n. It

Hence the j t h term in (2.8.13) equals

All summands in (2.8.13) except Q(n-k,lk)(x; q, t) cancel out, and (2.8.11) follows. Applying the automorphism w,,~and using the duality theorem (5.2) of [139] and Proposition 6.2, we obtain (2.8.10).

The theorem has several interesting specializations; for example, for the HallLittlewood functions (q = 0), monomial symmetric functions (q = 0, t = l ) ,etc. For q = t, formulas (2.8.10), (2.8.15) reduce to (2.8.6), and formulas (2.8.11), (2.8.16) reduce to (2.8.7). For general q, t, and x = (xl ,x2,0, . . . ), (2.8.11) is the determinant of the representation (2.8.2) for Rogers polynomials. 8.4. If a diagram X is not a hook, the formulas become more complicated. For example, for X = (a 1,2, lb-l)

+

where (q - t) (1 - ta) (1 - qb-lt) (1 - qbta+l) (1 - t) (1- qta) (1 - qb-92) (1 - qbta-1) ' Recall the definition of the F'robenius parametrization of a diagram X E P. Denote by d = d(X) the largest i such that Xi 2 i, and let fi = Xi - i, gi = A', - i for i = 1 , 2 , .. . , d, where A' stands for the conjugate partition of A. Then the Frobenius symbol ):; uniquely determines A.

K=

(i:::::

59. BRANCHING O F MACDONALD POLYNOMIALS

105

+

Denote by f&g = (f 1, l g ) the hook diagram with arm length f and leg length g. There is a well-known expression for Schur functions in terms of hook Schur functions sfhg(x):

CONJECTURE. For all X

E P there exists a rational function MA = h'fA(q,t)

such that

for all x = (x1,x2,.. .). For example, for X

and for X

=

(a

=

(a, b)

+ 1, 2, lb-')

59. Branching of Macdonald polynomials 9.1. The classical Hall-Littlewood polynomials have numerous applications in representation theory ([49], [143]). For example, consider GHL-functions X i , Yk in the Hall case (H) as t 4 -1. Then the characters of projective representations of the symmetric group B,, found by Schur, can be written as

where X is a strict partition of n with m parts; &(A)= 0 if ( n - m) is even, and &(A)= 1 otherwise. Here p ranges over all partitions with odd parts which index the conjugacy classes of the second kind in 6,. The second example is the Hecke algebra H,(q) of the symmetric group 6,. It was shown in [19] that the characters of H,(q) can be described in terms of symmetric functions by a formula similar to Frobenius' formula:

where P = (pi,p2,. . .), rp = rplr p z . .. , and r, = r, (x; q) = qm-'

. P(,) (x; 0, l/q).

We denote by X; the values of the irreducible characters xX,X E P,, on the elements of the form Cp

where t l , . . . ,t,-l

= (tl...tpl-l)(tpl+l...tP1+PS-l).

are the standard generators in H,(q).

. .. ,

2. BOUNDARY O F T H E YOUNG GRAPH AND MACDONALD POLYNOMIALS

106

9.2. Another point of view on symmetric functions arises from the description of the characters of factor representations of the infinite-dimensional algebras @[6,]and H,(q). According to Thoma's result [161], such a character x,,~is indexed by a pair of nonnegative sequences

>

>

>

> P2 2 . . . 0, and y = 1 Cai Cpi > 0. > 2, equals

such that a1 aa ... 0, ,Bl value of x,,p on the cycle c(,), m

-

-

The

A similar result was obtained in [19] for the Hecke algebras H,(q):

where P(,)(a,p; 0,114) is defined by substituting pn = p n ( a , P) into (2.5.2). For the so-called Markovian traces trq,t of the algebra H, (q) which were studied in connection with an algebraic construction of Jones' knots invariants [114], we have

where the coefficients are related to Macdonald HL-polynomials as follows:

If b E Hn(q) c H,(q), then the sum in (2.9.2) is over X E P N for any N 2 n. The scalar product (2.5.1) and the corresponding GHL-functions in the case (M) can be related t o representations of quantum groups and their centralizer algebras, see [148].

6,

9.3. The description of the characters of factor representations of the group = l q 6, reduces to the following question:

(*)

Find all homomorphisms cp : A

-+

@ such that cp(sx)

> 0 for all X E P.

A homomorphism cp can be defined by the generating function

Then the answer to (*) is as follows:

where a , p are the same as in (2.9.1). Consider a more general problem:

(**) Find all homomorphisms cp : A where PA= PA(x; q, t).

-+

@ such that cp(Px)

> 0 for all X E P,

59. BRANCHING O F MACDONALD POLYNOMIALS

107

CONJECTURE. T h e generating functions (2.9.3) for homomorphisms that are positive i n the sense of (**) are of the form

where

One can show that all these homomorphisms are indeed positive. In some cases this list is known to be conlplete: 1. in the Schur case (S) this was proved in [161, 131. 2. for projective characters of the group 6, (q = 0, t = -1) the conjecture was proved by Nazarov [51]. 3. in the case q = 0, t = 1 the conjecture follows from Kingman's result [130]; see also [34].

CHAPTER 3

The Plancherel Measure of the Symmetric Group In this chapter we study the properties of the Plancherel measure of the infinite symmetric group. For the sake of completeness, in $1we reproduce the results of A. M. Vershik and the author [12, 181 on the typical limit shape of large Young diagrams. These results were the origin of a considerable part of this work. In $2 we obtain an analogue of the central limit theorem for the characters of 6, which is a refinement of the main theorem of the first section. In 53 the theorem on the limit shape is used t o study the statistics of symmetry types of random tensors. In $54 and 5 we consider a q-analogue of the hook walk algorithm, and in 56 we compute the multivariate Selberg integrals as generalized Poisson integrals for appropriate branching graphs. 51. The typical shape of random Young diagrams

Vershik and the author [18] obtained order-sharp two-sided bounds on the typical and maximum dimensions of irreducible representations of the symmetric group G N as N -+ CO. Both problems were solved simultaneously and related to the theorem on the limit shape of typical Young diagrams which had been proved earlier in [12]. Since these theorems provide the motivation for many results of this book, we will reproduce their statements and sketch the proofs. 1.1. The main results. Let 6~ be the symmetric group of order N and denote by bN E yN the set of all equivalence classes of its complex irreducible representations. Given A E &iN, denote by dimA the dimension of A. The problem of computing the maximum dimension dim(A) was posed long ago (see [141]).Recall that by Burnside's formula

dim2 A = N!; A E ~ N

eN,

hence dimA < n ! for all A E and the natural normalization of dimension is d i m ~ l n ! .It was conjectured that there exist gigantic representations with d i m A / n ! 1/N. This conjecture was based on the numerical data obtained in [141] for N 5 75 (!); later it turned out that it fails for N = 81. However, one could think that m a x d i m h l f i ! 2 P ( N ) - l , where P is a polynomial. The next theorem disproves this conjecture and shows that the quotient m a x ~ dim A I f i ! decreases as e x p ( - c n ) , i.e., faster than any polynomial.

>

3.

110

T H E PLANCHEREL MEASURE OF T H E SYMMETRIC GROUP

THEOREM A. There exist positive constants c0 and cl such that, for all N = 1,2, ...,

The problem of computing the maximum dimension turned out to be very closely related to another problem, that of the typical dimension. Given A E kN, put pN(A) = dim2 AIN?. It follows from Burnside's formula that p~ is a probability measure on kN;it should be called the Plancherel measure. Note that the Plancherel measure is naturally distinguished; p,(A) is the relative dimension of the isotypic component of the representation A E E N in the regular representation of e N . It is with respect to this measure that statistics and asymptotics of characters should be studied. It turns out that the asymptotics of the typical (with respect to the Plancherel measure) dimension coincides in order with the asymptotics of the maximum dimension.

THEOREM B. There exist positive constants cb, ci such that N-oo

A : cb
1.8. The proof goes essentially along the following lines: we transform the formula for the dimension, then solve a variational problem, and prove that the solution is unique. For this purpose, the following expression for the hook integral was used in (121:

1 f 1'

= -//log21s

-t

. f f ( s ) . f l ( t ) dsdt.

These considerations allowed one to investigate the asymptotics of the shape and dimension of the typical diagram with respect to the Plancherel measure, but gave no information on the maximum dimension. A substantially new idea was to write the quadratic part of the hook integral in the form

11 f 1 '

=

// (f

-

s-t

(t))

' ds dt.

The appearance of the Sobolev norm and Hilbert integral in a combinatorial problem concerning Young diagrams of maximum dimension looks surprising. The proof of Ulam's conjecture was announced in [12] as an application of the theorem on the limit shape. It requires some new considerations compared with Theorem B; namely, we need an additional upper bound on the length of the first row of the random Young diagram. We give a detailed proof in Section 1.4. 1.2. The hook integral. The proof of Theorems A and B is based on the remarkable hook-length formula of Frame, Robinson, and Thrall [92]: dimh

N!

=-

n hi,

'

Here the product is over all squares (i,j) of the diagram A, and hij = Ai+Al-i-j+l (where is the length of the j t h column) is the number of squares in the hook with vertex (i,j ) . This formula is equivalent to the well-known Frobenius formula, but it is much more convenient because of its symmetry and multiplicativity. It implies the following formula for the Plancherel measure:

Xi

Taking logarithms, dividing by

where JA = 1

fl,and applying Stirling's formula yields

+ $ Czjlog d'z and

depends only on N

112

3.

THE PLANCHEREL MEASURE OF THE SYMMETRIC GROUP

Let y = F ( x ) be a compactly supported bounded nonincreasing function dey). Denote by h ~ ( xy), = F ( x ) fined on [O, w ) . Set F-' (Y) = inf{F(x) F-l(y)- x - y the hook at the point (x, y), and let SF = {(x,y) : 0 y < F ( x ) , 0 5 x < w) be the subgraph of F . The hook integral of the function F was defined in [la]as the double integral


0

PROOF. The Robinson-Shensted-Knuth algorithm (RSK) (see [152, 157, 133, 1271) defines a map from the space of sequences into the space of tableaux T. It maps a product measure with a continuous factor to the Plancherel measure. Moreover, by a well-known property of the RSK-algorithm, the length of the longest increasing subsequence coincides with the length of the first row of the corresponding Young diagram (Shensted's theorem, see [157]). The corollary follows now from Theorem D. Shensted's theorem admits a generalization: the number of squares in the first k rows of a Young diagram is equal to the maximum number of elements in the union of k monotonically increasing subsequences. This fact leads to a strengthening of Theorem D, obtained in [la],which we omit here. 52. Gaussian limit for the Plancherel measure In this section we show that the Plancherel measures of the symmetric groups + cc to a Gaussian random process in an infinitedimensional linear space. With respect to the law of large numbers for these measures, which was found earlier in [12, 181, this statement is an analogue of the central limit theorem. The idea of such a theorem was discussed in [138].

6, weakly converge as n

A

2.1. Main results. Let 6, be the symmetric group of degree n , and let 6, be the set of (equivalence classes of) its irreducible representations; the character of a representation X E 6, will be denoted by x'. Given a partition p = (1'12'2 . . . ) of n , consider the class of conjugate permutations in 6, having rk cycles of length k for k = 1 , 2 , . . . , and denote by the value of the character XX on this class. A

Xt

$2. GAUSSIAN LIMIT FOR T H E P L A N C H E R E L MEASURE

117

A

Let dim X = X&,) be the dimension of a representation X E 6,. By Burnside's theorem Ex dim2 X = n!, so that the weights Mn(X) = dim2 Xln! form a probability distribution Mn on the set g,, called the Plancherel measure of the group 6,. Define the random variable

as the (normalized) value of characters of representations X on the class of permutations with a single nontrivial cycle of length k. The main result of this section is the following:

THEOREM 1. For every x2,x3,. . . , x, (3.2.2) lim Mn{X E 6 , : pk(X)< x i , 2

n-00

E R there exists the limit

< k < rn) = k=2

In other words, the functionals cpk(X)are asymptotically independent and have Gaussian limit distributions with zero mean and variance k . Let us give alternative statements of Theorem 1 related to other systems of basic functionals. We will use the standard identification of the set 6, with the set y, of Young diagrams with n squares. Introduce the following notation for the unital Chebyshev polynomials of the first and second kinds on the interval [-2,2]:

+

sin(r 1 ) O sine '

t T ( 2 c o s 8 ) = 2 c o s r 0 , u,(2cosO)=

Let d be the length of the main diagonal of a Young diagram X = (A1, X2,. . . ) E Yn, and denote by iii = (Xi - i)/,,h, bi = (Xi - i ) / f i , i = 1 , . . . , d, its normalized Frobenius coordinates. For r = 2 , 3 , . . . , set (3.2.4)

tT(x)=

x

+

(tT(iii) (-I),+'

tT(ii)).

Z

We will denote by (f), = CxEyn f (A) Mn(X) the mean value of a function f with respect to the Plancherel measure Mn. THEOREM 2. The Plancherel mean value of the random variable (3.2.4) vanishes for even r and equals

f o r r = 2m+ 1. As n i a, the centralized variables :,(A) = t,(X) - (t,),, X E yn, are asymptotically independent and normal. The limiting variance t,(X) is equal to lim(;:),

= r.

Let c(i, j) = ( j - i)/,,h be the normalized contents of the square (i, j) of a Young diagram A. For r = 1,2, . . . , set ur(X) =

1 -

C

Jii ( i . j ) E *

ur(,).

j-i

118

3.

T H E P L A N C H E R E L M E A S U R E OF T H E S Y M M E T R I C G R O U P

THEOREM 3. The Plancherel mean value of the random variable (3.2.6) vanishes for odd r and equals

for r = 2m. A s n + cm, the centralized variables &(A) = u,(A) - ( ~ r ) n A, E Yn, are asymptotically independent and normal. The limiting variance ur(A) is equal to lim(gT)n = l / ( r 1).

+

Let us now regard Young diagrams as piecewise linear continuous functions w with derivative f1 (this viewpoint goes back to [12]). For instance, the one-square diagram w E yl is identified with the function w(x) = max(lxl,2 - 1x1). It follows from the law of large numbers established in [12] that the Plancherel mean value

of a random diagram with n squares (scaled by ,/6 along both axes) uniformly converges to the function

Embed the set yn of Young diagrams with n squares into the space of functions by associating with a diagram w E y, its-deviation Aw(x)from the curve (3.2.8); more precisely,

Identify the Plancherel measure Mm with its image under this embedding.

THEOREM 4. AS n + cm, the measures M, weakly converge to a Gaussian random process o n the interval [-2,2]. The limiting process has zero mean and covariance function

Substituting x = 2 cos cp, y = 2 cos $ and using the definition (3.2.3) of Chebyshev polynomials, we can write the function (3.2.10) more explicitly: B ( 2 cos cp, 2 cos $) =

1

2.2. Proof of Theorem 1. Let y = Un>oyn be the set of all Young diagrams (or, equivalently, of all partitions of positive integers). With each partition p E y we associate the family of central elements c,;, E Q[G,], n 0, defined by

>

-

Here p = (Ir1,2rZ,.. . ) is a partition of the number r = Ipl = k r k , C(p,ln-r) is the uniform probability distribution on the conjugacy class C(p,ln-r) of G,, and (n), = n ( n - 1) . . . ( n - r 1). For n < r we set c,;, = 0. It is remarkable that in the basis (3.2.12) the convolution is stable in n:

+

$2. GAUSSIAN L I M I T F O R T H E P L A N C H E R E L M E A S U R E

119

PROPOSITION 1. For each pair of partitions a, T there exists a (unique) finite family of integral nonnegative coeficients f$,, such that

for all n

> 0.

Without reproducing the proof, we restrict ourselves to a clarifying example: C(2);n

* c(2);n= c(22);n+ 4~(3);n+ 2 ~ ( 1 2 ) ; n

In the basis of conjugacy classes the coefficients of expansion depend on n:

Define the class ring C as the free abelian group with generators {cp), p E and the multiplication determined by the structural constants f$,,. The elements cp E C determine functionals on G = Un,o Gn by the formula cp(A) = X X ( ~ p ; ndim ) / A. It is easy to see that

y,

A

(3.2.14)

(c,

Let deg(cp) = ri

* cr ) (A) = C,

+ /pi. If f:,,

(A) C, (A),

# 0 in

A

AE

6.

(3.2.13), then

Thus we have defined a filtration on the class ring; we will need a detailed description of the corresponding graded ring gr C. By definition, this ring is additively generated by the same basis {c,}, and the multiplication is defined by the highest degree terms: deg p=deg a+deg r

In the formulas below, products and powers of elements cp are taken in the ring grC, i.e., up to terms of lower degree. The elements ck = c(k) corresponding to partitions p = (k) with a single part will be called the fundamental classes.

PROPOSITION 2. The elements cl, c2,. . . are algebraically independent and generate the ring gr C Z[cl, ca, . . . 1. If partitions a, T have no equal parts of length greater than one, then c, c, = c,",, where a U T is the union of the parts of a and T . If all parts of a partition are equal to k, then

--

(3.2.17)

C(kr)

where H r ( s ) =

=

SL H r ( ~ k / ~ k ) ,

ez2l2( e - ~ ' / ~ ) ( ' )are the Hermite polynomials and sk

The proof is based on the identity

where

r~

is assumed to have no parts of length k. By induction we obtain

=

@.

120

3.

T H E PLANCHEREL MEASURE OF T H E SYMMETRIC GROUP

in the ring gr C. The irlversiorl of this formula gives (3.2.17). Let us show that the Plancherel mean values (f), for f E C are polynomials in n. LEMMA.

(Cpf p ~ p ) n= C

k

f(lk)

. (n)k.

The proof follows from the orthogonality relations for characters of the symmetric group:

It follows from this lemma and (3.2.19) that

and (ci), = 0 for odd r ; moreover,

) / ~ p E Y ; i r ~particular, the variables pk = p(k) are Let pp(X) = n ~ ~ " g ( " pcp(X), equivalent to those defined by (3.2.1). Denote by ( p y p y . . . jCx3= l i m ( p 7 p T . . . ), the limiting momerlts of the Plancherel measure. Orle can see from (3.2.20), (3.2.21) that

Icm (2m - l)!! if r

= 2m,

ifr=2m-1, and

Thus the momerlts (3.2.22) coincide with the moments of the normal distribution with variance k , and (3.2.23) implies that the random variables p2,p3,. . . are independent with respect t o the limiting measure. Theorem 1 is proved. 2.3. Proofs of Theorems 2-4. Consider the modified F'robenius coordinates defined by ai = Xi - i 112, bi = X: - i 112. According to Lemma 1 from [13], the supersymmetric Newton sums

+

+

in the coordinates ai, bi of a Young diagram X belong t o Q[cl, c2, . . . ] and generate it as a ring. It is easy to check that the Newton sums

in the contents of squares of X generate the same ring.

53. DISTRIBUTION O F SYMMETRY TYPES O F HIGH RANK TENSORS

121

PROPOSITION 1. The normalized fundamental classes (3.2.1) can be expressed i n t e r n s of the sums (3.2.24) as follows:

where 62m-1

= 0,

+

62m = ( - l ) m / m ( m I ) , and o(1) refers t o the L2-norm.

Let us compare (3.2.26) with the explicit formula for the Chebyshev polynomials of the first kind:

Theorem 2 follows now from Theorem I. Theorem 3 is equivalent t o Theorem 2, since p,+l = ( r 1)T, up t o terms of lower degree deg. In order t o prove Theorem 4 , observe that the expression

+

is an integral sum for the double integral

'S

u,(x) d x d y = 2 where that

u,(x) ( G ( x ) - 1x1) d x ,

Dw= { ( x ,y ) : 1x1 5 y 5 G ( x ) ) and G ( X )= w ( x f i ) / f i . It is easy t o check

Therefore

where A,(x) is defined by (3.2.9). Theorem 4 now follows from Theorem 3. 53. Distribution of symmetry types of high rank tensors = The symmetric group G N acts in the space C m of rank N tensors over m-dimensional space by permutations of factors. Denote by yN the set of all Young diagrams with N squares, and by yN,,, the subset of diagrams with a t most m rows. The space decomposes into the direct sum of primary components under the act ion of G :

V N ,=~

@

VN,m(A).

A ~ Y N , ~

Tensors from V N , ~ ( Aare ) said to have symmetry type A. Denote by ~ N , ~ ( =A dim ) VN,,(A)/ dim VN,, the relative dimension of the primary component VN,,( A ) . The numbers pN,m ( A ) determine a probability measure on the distribution of symmetry types. The purpose of this section is to describe the asymptotic behaviour of p ~ , ,as N -+ cc in two cases: when m .v N / y , y = const, and when m = const.

$3. DISTRIBUTION OF SYMMETRY TYPES OF HIGH RANK TENSORS

123

PROOF. Let A' be the Young diagram with ( N - XI) squares obtained by removing the first row from A. Clearly, dim X 5 dim A', and hence

(E)

pN(h)= Let p

= X i IN,

9

5 (N)

N!

X i Xi!(N - Xi)!

q = 1 - Xi/N; the inequality

.-

< 1 implies

(E)pXlqN-X1 -

i.e., -q log q 5 p. Using the bound XI! 2 (Xl/e)'l and taking logarithms, we obtain


0, < E)

=

1.

PROOF. Consider the sets of diagrams

DN(c) = {A E

YN : ~ N ( X2) exp - c f i ) ,

where c E R. It follows from the well-known Hardy-Ramanujan bound p(n) = o(exp 2 x m ) on the number p(n) p~,,-measure:

=

lYN 1 of all partitions that D N , is~ of asymptotically full

pN,m ( Y N ,\~V N , ~ ) 0 as N Let us show that VN,mc VN(C)for sufficiently large c. +

+

0.

$3. DISTRIBUTION OF SYMMETRY TYPES OF HIGH R A N K TENSORS

125

THEOREM 2. AS N 4 cc (with m fixed), the measures P N , weakly ~ converge to the absolutely continuous measure Pm i n the cone Cm with density

where

is a normalization constant.

PROOF.Let be the joint distribution in Hm of the reduced row lengths 21,. . . , xm with respect to the multinomial distribution

By the Moivre-Laplace theorem ([25], $13) the measure the Gaussian distribution in Hm with density

weakly converges to

The formulas of Section 1 yield

-

1 1 . .. ( m - I)!

ncXi

-

xj)2~(m-l)m/2

(N/m)(m-l)m/2

Theorem 2 follows. The measure Pm arises in various problems (see [149, 1421). It is easy to obtain the most probable values of reduced row lengths: LEMMA4. The function cpm(x) attains its maximum value i n the cone Cm at -

the vector z = J $ ( z l , . . . , z m ) , where polynomial Hm ( 2 ) .

21,.

. . , z m are the zeros of the Hermite

The proof reproduces the proof of Stieltjes' theorem, see [56], Section 6.7. COROLLARY 2. Assume that the dimension m of the ground space is fixed and let N -+ co. The space VN,,(X) of tensors of symmetry type X has the maximum

where z l , . . . , zm are the zeros of the Hermite polynomial

For the finite-row analogue of the Plancherel measure a close result was obtained in [68].

126

3.

T H E P L A N C H E R E L MEASURE O F T H E SYMMETRIC G R O U P

$4. A q-analogue of the hook walk algorithm Greene, Nijenhuis, and Wilf [102], [103] suggested a remarkable probabilistic algorithm, the hook walk. This algorithnl was used for two purposes: for proving the hook-length formula and for generating randorn Young tableaux distributed according to the Plancherel measure. In this section we present new applications of the original Greene Nijenhuis Wilf algorithm and describe a y-analogue of this algorithm arid two of its applications. The first application is related to an interesting family of distributions on (infinite) Young tableaux which should be regarded as a natural y-deformation of the Plancherel measure. These distributions are in a one-to-one correspondence with Markovian traces on the Hecke algebra which arise in one construction of Jones' invariant for topological knots arid links. The new algorithm allows one to generate raridonl Young tableaux with these distributions efficiently. The second application is related t o a new interpretation of the kriown qanalogue of the hook-length formula. It turns out that one can assign rnultiplicities (rational functions in q) to the edges of the Young graph so that the recursively defined "y-dimension" of the Young diagram will be given by the q-hook-length formula. For y = 1 all rnultiplicities are equal to one, arid the proof yields the classical formula. Let us reproduce some kriown facts concerriirig the hook-length formula and the Plancherel measure, using the riotatiori and ternlinology of [49]. 4.1. The hook-length formula and transition probabilities. Given two Yourig diagranls A, A, we write X /" A if X c A arid IAI = ( X I 1, i.e., A has exactly orie square rnore than A. The relation X /" A provides the set y of all Young diagrams with the structure of a directed graph (the edges of this graph being pairs (A, A) with X /' A). Directed paths of this Young graph exiting from the initial vertex A = 8 are called Young tableaux. The dirnensior~of a Yourig diagram X is the number fx defined recursively as follows: fo = 0 for the enlpty diagrarn X = 0, arid

+

It is clear from the definition that fA is the number of Young tableaux of shape A. It is also knowri that fx coincides with the dimerisiori of the corresporiding representation of the synlrnetric group. According to the classical hook-length fornlula [92], fx = n!/ h i j , where hij is the hook length of the diagram X at the square ( i ,j ) . In this context, the Plancherel measure is a Markov probability meamire, on the space 7 of infinite Young tableaux, with transition probabilities

n

where n = IXI is the number of squares of A. Using the hook-length formula, we cau rewrite this expression in the form

34. A q-ANALOGUE O F T H E HOOK WALK ALGORITHM

127

Let us explain our notation. Let M be a point in a diagram X (regarded as a part of R): with coordinates i , j . The number c(M) = j - i is called the contents of M . Denote by MI, M2, ...,Md the successive vertices of the internal corners of A, and by Nl, N2, ...,Nd-1, the vertices of its external corners lying between them. In (3.4.2), xk = c(Mk), yk = c(Nk) are the contents of the vertices of the south-east border of X (see Figure 19). We write Rk in place of RAAif the square that distinguishes A from X is attached to Mk. In what follows, we will need to inscribe X into the rectangle with vertices No, Nd; denote by yo, yd their contents.

FIGURE19. For the definition of cotransitional probabilities. It follows from the well-known formula (n

X

+ I )f x = C A r x ffAAthat

Let be the complement of X in the rectangle (Figure 19). Using the hooklength formula, we can write the recurrence relation (3.4.1) for the diagrams A /" in a form similar to (3.4.3):

X

- yi-l)(yj - x j ) . Conversely, in order to prove the hookwhere S = xl,i<j 0 (see [17], [19]). The following description of these measures is well-known. The ergodic central measures are indexed by pairs of nonincreasing sequences a = {ai)gl, /3 = {Pi)zl of nonnegative numbers satisfying the condition

The transition probabilities of the measure corresponding to a pair a,P are given by the formula

where the extended Schur functions s x ( a ;P) are defined, for example, by F'robenius' formula ([49], 1.7.6)

xi

+

>

(-l)n+l /?: for all n 2. The where p l ( a ; p) = 1 and p,(a;P) = Cia: Plancherel measure corresponds to the zero sequences. Expressions of the form W(r, q) = (1 - qT+l)/(l q ) ( l - qT)play an important role in the description of irreducible representations of Hecke algebras (see [33]). Given a triple of diagrams p /" X /" A, let N, and MI, be the vertices to which the squares X\p and A\X, respectively, are attached. The number r p x A = MI,) c(N,) = xk - ,y is called the axzal distance.

+

$4. A q-ANALOGUE OF T H E H O O K WALK A L G O R I T H M

129

DEFINITION.A central measure P is said to be a knot measure if there exists a number q such that the sun1

does not depend on the pair p

/" A.

The choice of the term is due to the fact that the characters of H , ( q ) associated with knot nleasures are used for constructing topological invariants of links in IR3. See [19]for a simple description of knot ergodic central measures. Let us enumerate the parameters of these measures using the notation [n] = ( I - q n ) / ( l- q ) .

EXAMPLE 1 . Let a

= ( ( 1 - q ) q k } E 0 ,P = 0 .

EXAMPLE 2. Asslime that

0

= 0, P =

In this case

( ( 1 - q)qk)r=,. Then

EXAMPLE 3. Let 0 < q < 1, 0 = { q k ( l - q ) / ( l - q r n ) } T ~Pt ,= 0 .

EXAMPLE 4. Let

0 =

0,

=

{ q k ( l - q ) / ( l - q r n ) } ~ In ! ~this . case

x,?,

By definition ( [ 4 9 ] ,1.1.5),n(p) = (i - l ) p i for a diagram ( p l , p 2 , . . . ). The first two examples are special cases (when t = 0 and t = 1) of the following twoparameter family of ergodic knot central distributions.

EXAMPLE 5. Asslime that 0

m - 1, are defined by the formula

+

+

These probabilities are well-defined, as may be seen by computing the moments of the transition probabilities of the Plancherel measure, see (4.1.18):

The Markov chain with transition probabilities (3.6.15) is central, since the probabilities of all Young tableaux of shape X coincide and are equal t o M ~ ) ( x/ )d(X), where

The original formulas (3.6.12) and (3.6.14) are obtained from (3.6.15) and (3.6.17) by letting v + m.

140

3.

T H E PLANCHEREL MEASURE O F T H E SYMMETRIC GROUP

Let us now find the Poisson integral representation for the central measures (3.6.15) on y(m). Recall that the radial embedding of the graph y(m) into its boundary A, is defined by the formula

PROPOSITION. Consider a sequence of diagrams A(") E limits

yn(m) such that the

exist. Then there also exists the limit

where

In other words, the weak limit of the discrete distributions i ( M P ) ) on Am exists and equals the absolutely continuous measure

PROOF.Using the well-known formula

1

nm

bEX

+

n i < , ( X i - Aj j - i ) = nj=,(xi+m-i)! ,

rewrite (3.6.17) in the form

n!

n

m

++

m

+

( m- i l ) x , M?) ( A ) = ( X i - ~ ~ + j - i ) ~ (xv n- i l ) ~ , ( ~ im - i ) ! +m-i)!. i=l l 0 and can be identified with the simplex (3.6.27) from Section 6.2. On the contrary, the Poisson kernel does depend on 8 and coincides with Jack polynomials: (3.6.31)

@(A;a ) = Px(a; Q), A E y ( m ) , a E A,.

Following the scheme of the previous section, we must now define an analogue of the transition probabilities (3.6.15) which depends on 8 and determines central Markov chains on the Jack graph ~ ( ' 1 . Remarkably, this can be done by using the following simple idea, which also provides a significant interpretation of the formal parameter 8. Let us consider, along with a Young diagram X E y, its scaled versions X(Q) obtained by preserving all horizontal dimensions (row lengths) and multiplying all vertical dimensions (column lengths) by Q. For example, if X = (72, 44, 12) and Q = 112, then X(1/2) = (7, 42, 1). In the general case, the rectangular diagram X(Q) is no longer a Young diagram; however, its minima xk(Q)and maxima yk(Q)are in a one-to-one correspondence with the minima and maxima xk, yk of the original Young diagram A. Notice that the contents of a square (or, better to say, of a point) with coordinates i, j in the scaled diagram X(Q) should be defined by

For 8 = 1 we recover the ordinary contents. The contents of the external and internal corner squares of the scaled diagram X(Q) will be denoted by xk(Q)and yk (Q),respectively. Let us define the transition probabilities by the same formula (3.6.15), but applied not to a Young diagram X but rather to its scaled version X(Q). This is possible, since due to Theorem 32 we have the definition of transition probabilities for any rectangular diagrams. So set

(these are the Plancherel transition probabilities for the scaled diagram X(Q))and let

where k = 1 , 2 , . . . , d runs over the numbers of external corners of A. It follows immediately from the general lemma, which claims that the first three moments of the transition distribution do not depend on the shape of the rectangular diagram, that (3.6.34) defines a probability distribution. Using formula (3.6.29) for the edge multiplicities of the graph ~ ( ' 1 , one can easily verify that the hlarkov chain with transition probabilities (3.6.34) is central

56. M U L T I P L E S E L B E R G I N T E G R A L S

143

on the corresponding truncated graph ~ ( ' ) ( m )The . distribution of the state at the nth leiel Jn(0) ( m )is given by

where ( x ) , = x ( x The limit as n

+ 1 ) . . . ( x + n - 1 ) is the Pochhammer symbol. -t

oo can be obtained as in (3.6.20), above.

PROPOSITION. Let v = A + ( m - 1)d, A > 0 , and assume that for a sequence of Young diagrams the limits (3.6.35)

lim

n-cc

A:"' n

j = 1 , 2 , . . . , m,

-=aj,

exist. Then lim n m l M ; ~ ' * ) ( A ( ~ =)C) m ( 0 )

(3.6.36)

n-cc

n

n m

lai - a j '1

l 0 such that w,(u) = lul for lul > c for all n = 1 , 2 , .. . .

$2. CONTINUOUS DIAGRAMS AND R-FUNCTIONS

151

A diagram w is called rectangular if it is piecewise linear and wl(u) = f1 for (almost) all u E W (see Figure 21, on p. 145). A rectangular diagram is uniquely determined by the coordinates of its minima X I , .. . , x, and maxima yl, . . . ,y,-1, which form a pair of interlacing sequences:

Conversely, each pair of interlacing sequences (4.2.3) uniquely determines the rectangular diagram w with the corresponding extrema and with centre a t the point z = C xk - C yk. The area of the rectangular diagram is

Note that the set Do of rectangular diagrams is dense in V. 2.2. The R-function of a diagram. Given a diagram w E V[a, b], consider the function a(u) = i(w(u) - Iuj). This function enjoys the following properties:

0'(u)

+ >

~ ' ( u ) 1 0 if wl(u) - 1 5 0 if max a ( u ) = w(0)/2,

=-

u u

< 0, > 0,

u

The differential da(u) = ul(u) du determines an absolutely continuous (signed) measure called the charge of the diagram; it is supported by the interval [a, b], and the total variation of the charge equals Jdal = w(0).

DEFINITION. The R-function of a diagram w E V[a, b] is given by (4.2.4)

2

1 Ru(u) = ;exp[

= - exp

u

:J,

-

d(w(t) - Itl) t-u .

This function is defined and holomorphic outside the interval [a, b].

EXAMPLE 1. Let w = 0 , where

then Ra(u) = (U - d-)/2

for u

>. 2.

EXAMPLE 2. Let w = w, be the diagram which is centered a t z < 1 and q = 1 - p) and linear in the interval [-I, 11, that is,

0 . 1.

Then da(t) = and R,(u) = (u

{qdu -pdu

if

-lPdefined by the same diagram w on different subintervals are related by the integral formula

which is equivalent to the defining equation (4.4.11). The theorem follows.

55. Differential model of growth of Young diagrams 5.1. The idea of a differential model. The starting point for this section is the result on the asymptotics of random growth of Young diagrams obtained in [12, 1381. Those papers dealt with a discrete time Markov process whose state space is the set of all Young diagrams. At each step of the process a new random square is attached to the current diagram, and the definition of transition probabilities is motivated by asymptotic problems of the representation theory of symmetric groups, see [17]. Keeping in mind the representation-theoretic origin of this process, we call it the Plancherel growth process. As a diagram grows, its area increases to infinity. If we uniformly scale the diagram so that the rescaled diagram has unit area, the edge of the diagram looks more and more like a continuous curve. The key point of the theorem discovered and proved in [12, 1381 is the following. In the course of the Plancherel growth process almost all Young diagrams (after the normalization of the area) become uniformly close to a common universal curve. In a natural coordinate system the equation of this curve is

Figure 8 on p. 35 shows the graph of this curve, along with a random diagram with a hundred squares obtained by a computer simulation of the Plancherel growth process. For fifteen years the appearance of the curve (4.5.1) seemed to be a specific isolated result. At present the situation is different: it turned out that the same curve arises naturally in various problems of one-dimensional mathematical physics and function theory (see [118], [38], [37]). As shown in [38],the curve (4.5.1) describes the common asymptotics of separation of roots for a wide class of orthogonal polynomials (including the classical polynomials of Chebyshev, Hermite, and Laguerre). The same curve describes the typical character of mutual separation of frequencies under a random linear constraint in a wide class of linear mechanical systems (see [38]). Below we establish new characterizations of the curve (4.5.1) which relate two contexts where it appears: the Plancherel growth of Young diagrams and one-dimensional problems of mathematical physics. The main idea is to replace the original discrete time random process by a continuous time deterministic process with the same asymptotic behaviour as t + OC).

Roughly speaking, the elimination of randomness is achieved as follows. Given a certain state of the Plancherel growth process, assume that the new square can be attached to the current Young diagram a t points X I , .. . ,x,, with probabilities p1,. . . , p,. Then, instead of randomly attaching the entire square to one of these

55. DIFFERENTIAL MODEL O F GROWTH O F YOUNG DIAGRAMS

169

points, we would like to attach simultaneously a part of the square proportional to the probability pk t o each point xk. In order to implement this idea we substantially extend the state space by considering along with Young diagrams the limiting curves of the form (4.5.1). See $2 for a precise definition of continuous diagrams. The extended space of diagrams V is an infinite-dimensional compact space which is canonically homeomorphic to the space of probability measures on an interval. The correspondence which associates with a diagram w E V a probability distribution pw is by no means trivial. It was implicitly used in connection with the Markov moment problem [47] (cf. [37]). However, in our context the map w H p, arises quite naturally: if w E Y is a Young diagram (maybe rescaled), then p, is its transition distribution in the Plancherel growth process. The correspondence w H pw can be uniquely extended by continuity to general diagrams w E V (see [I181and $3). Another interpretation of the correspondence w H pw is related to the partial fraction expansion ($2). The history of growth of a general diagram (an analogue of a Young tableau) is described by a curve w(.,t), 1 < t < oo, in the space 'D. The diagrams w(.,t) are assumed to increase (with respect t o the inclusion of subgraphs) with t. The infinitely thin layer between the graphs of the diagrams w(u, to) and w(u, to dt) determines a one-dimensional distribution with density CW:(U,to) (where C is a normalization constant); it is natural t o think of this distribution as a tangent vector to the tableau w(., t) a t the point t = to. Equating the tangent and transition distributions, we arrive at the basic dynamic equation

+

A more closed form of (4.5.2) is given in (4.5.10) below. Equation (4.5.2) means that the diagram w(., t) always grows in the direction of its transition measure. Thus we can expect that the asymptotic behaviour of solutions of (4.5.2) is related to that of the Plancherel growth. Our main result (the theorem of Section 5.6) completely justifies these expectations: the diagram R from (4.5.1) is the unique, up to scaling, fixed point of (4.5.2), and all other solutions are asymptotically attracted by this point. In the course of our analysis we rewrite (4.5.2) in several equivalent forms. Surprisingly, one of the equivalent restatements of (4.5.2) is the quasilinear firstorder equation

which describes the free motion of a one-dimensional medium of noninteracting particles. This is one of the simplest equations illustrating nonlinear phenomena, in particular, the appearance of shock waves. In terms of equation (4.5.3), the curve (4.5.1) corresponds to the automodel solution

where (4.5.4) It follows from the main theorem that if the initial wave R(x, 1) is analytic at infinity, then, in the course of its evolution controlled by equation (4.5.3), its shape

170

4.

YOUNG DIAGRAMS IN PROBLEMS OF ANALYSIS

in a neighbourhood of the point x = cc tends, as t + ca,to the shape of the automodel solution (4.5.4). The section is organized as follows. In Section 5.2 we define a continuous analogue of Young tableaux, and in Section 5.3 we introduce the differential equation which controls the growth of continuous diagrams. Then we prove three theorems characterizing the diagram R. I11 Section 5.4 we show that this diagram determines the unique automodel solution of the basic equation. According to Section 5.6, all other solutions are asymptotically attracted by this solution. In Section 5.5 we claim that R is the unique (up to scaling) diagram whose transition distribution coincides with the radial distribution (defined in Section 5.5 from simple geometric considerations). 5.2. Continuous tableaux. Like Young diagrams, general continuous diagrams can be ordered by inclusion of subgraphs: wl + w2 if D,, c D,,. A tableau is a continuous D-valued map defined on some interval [to,m) and strictly increasing on this interval. Thus a tableau is a family of diagrams w(.,t) increasing in t. All these diagrams have a common centre z(t) = z ; without loss of generality we assume that z = 0. The area A(t) continuously increases with t ; it will be convenient to take it as a parameter, assuming that A(t) = t. Given a tableau w(u, t ) , denote by u(u, t) = (w(u, t) - 1u1)/2 the corresponding charges (see Section 2.2). Taking into account our assumptions on the parameter t and the normalization of the area, we conclude that

DEFINITION. The function T ( u , t ) = u:(u,t) is called the tangent density of - at the point w(., t). the tableau {w(.,t))t>to Differentiating (4.5.5) in t, we obtain that T(u, t) is indeed the density of a probability distribution, which we call the tangent distribution: (4.5.6)

J

T(u, t ) du = At(t) = 1.

Let us compute the moments of the tangent distribution. LEMMA.Let pn(t) be the moments of a diagram w(u, t ) in a tableau {w(., t))t>to. Then the moments of the tangent density can be written in the form

PROOF.By the definition of the tangent density,

and the lemma follows from the equations

55. DIFFERENTIAL MODEL O F GROWTH O F YOUNG DIAGRAMS

171

5.3. The basic dynamic equation. Consider a growing diagram, that is, a tableau {w(.,t)It2t,, and let pt = P,(.,~)be the transition measure of the diagram w(u, t). We require that the diagram w(u, t ) grow in the direction of its transition distribution:

Relation (4.5.8) should be regarded as a n evolutionary equation in the infinitedimensional phase space of diagrams V ,where the correspondence w H p, determines the canonical vector field (more precisely, the field is defined on the subset of diagrams with absolutely continuous transition measures). Using the definition of the transition measure, we can rewrite (4.5.8) in a closed form: cr;(u, t) du = exp for sufficiently large x. 1 - u/x Let us give two other forms of the basic dynamic equation (4.5.8): in terms of the moments pn = pn(w) of the diagram and in terms of the generating function

J

(for a fixed t this function coincides with the R-function from 53, formula (4.3.3)). PROPOSITION. The following forms of the basic dynamic equation are equivalent:

PROOF. The equivalence of (4.5.10) and (4.5.11) follows from the lemma of Section 5.2. Consider the series

Since pl (t) = z = 0, (4.3.5) implies S ( x ,t ) = log(xR(x,t)), whence Si(x, t) = R-'(x, t ) R:(x, t). Comparing the series

we see that (4.5.11) and (4.5.12) are equivalent too. Note that (4.5.12) is one of the simplest examples of quasilinear first order partial differential equations; among other applications, it is used as an illustration of

4.

172

YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS

nonlinear phenomena such as breaking of wave crests or the birth of shock waves. In one of its numerous interpretations, (4.5.12) is regarded as the equation describing the free motion of a medium of noninteracting particles. In this case R(x, t ) is thought of as the velocity of the particle located a t point x at time t.

5.4. A u t o m o d e l solutions. For every diagram w of unit area A(w) = 1, the formula

determines a tableau. Tableaux of this form will be called automodel. Let us show that the unique automodel solution of the basic dynamic equation is generated by the diagram R. THEOREM.Let w(u) be an arbitrary diagram of unit area A(w) = 1, and w(u,t) = \/tw(u/\h), the corresponding automodel tableau. If its charge o ( u , t ) satisjies (4.5.10), then w(u) = R(u).

PROOF.Consider (4.5.10) in the equivalent form (4.5.12). It is easy to check that the moments p,(t) of the diagram W(U,t) can be expressed in terms of the moments p, of the initial diagram w(u) in the form

By the homogeneity of formulas (4.1.17), we have a similar relatiorl for the moments of the transition measures: h,(t) = tn/2h,. Thus the function (4.5.9) is equal to R(x, t ) = R(x/&)/&. Substituting this expression in (4.5.12) yields, after a simple calculation, the relatiorl (xR(x))' = (R2(2))' ; hence xR(x) and R2(x) differ by a constant. Comparing the constant terms, we corlclude that R2(x)- xR(x) 1 = 0. The only solutiorl of this equation vanishing as x + +co coincides with the generating function of the diagram R. The theorem follows.

+

Irlformally speaking, we have shown that R(u) is the only diagram that remains self-similar (i.e., does not change its shape up to rescaling) in the course of the growth in the direction of its ow11 transition measure. For automodel tableaux we have

The expressiorl in the right-hand side is determined by the diagram w(u) = w(u, t) for a fixed value o f t ; it has a simple geometric interpretation which will be discussed in the next section.

5.5. T h e radial d i s t r i b u t i o n of a diagram. Given a pair of arguments a < p, denote by Ma, Mp the corresponding points of the graph of a diagram w, and let A,P be the area of the curvilinear sector in between the rays OM,, OMo (where 0 is the origin), and the arc M,Mo of the graph (see Figure 24). 1

A""

P (W (u) -

w l ( u ) ) da =

(o(u) - uol(u)) du.

55. DIFFERENTIAL MODEL O F GROWTH O F YOUNG DIAGRAMS

173

FIGURE24. Radial distribution of a diagram. PROOF. Consider the ray exiting the origin 0 a t an angle 0 and meeting the graph of the diagram at a point M. Denote by r ( 0 ) the length of the segment O M , and by u the abscissa of M ; u and 0 uniquely determine each other: 0 = B ( u ) . Integrating in polar coordinates yields Aa8 = r 2 ( 0 )dB. Substituting 0 = 0 ( u ) gives ctg B = u / w ( u ) , whence

-a

[~(u )u w l ( u ) ]d u

= w 2 ( u )d ( u / w ( u ) )=

~,e( $

- w 2 ( u ) sin-2 0 d0 = - r 2 ( 0 ) dB.

The second claim of the lemma follows immediately from the definition of the charge of a diagram. Let us define the radial density of a diagram w E

D by the formula

It follows from the lemma that

so that the radial density determines a probability distribution. We call it the radial distribution.

PROPOSITION. Let pn be the m o m e n t s of a diagram w E D. T h e n the m o m e n t s of its radial distribution are equal t o

PROOF. We have

As a moment generating function for the radial distribution we may take (assuming pl = z = 0 ) the function

174

4.

YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS

THEOREM. Assume that the transition distribution of a diagram w coincides with its radial distribution, dp,(u) = p,(u) du. Then w coincides, up to similarity, with the diagram R for all u E R:

where A = A(w).

PROOF.Let us express the equality of moments of the distributions under con= h,, in terms of the function R(x) = hn x-("+l). sideration, ~ , + ~ / 2 ( n + l ) A By (4.5.17) we have

En,,

(since xR(x) = exp S(x)). Comparing with the expansion

+

yields (xR)'/R = 2AR1, or (xR)' = (AR2)'. Thus AR2 - XR C = 0 for an appropriate constant C. Comparing the lowest terms of the expansions xR(x) = 1

+ t/x2 + . . .

and

AR'(X)

= A/x2

+ . .. ,

we find that C = 1, so that

for x Finally, R(x) = (x - d=)/2A remains to observe that the function

> 2 a (since R(x) + 0 as x + + m ) . It

coincides with the analogous generating function in the right-hand side of (4.5.18). Note that according to (4.5.14) the tangent distributions of an automodel tableau coincide, at each diagram, with its radial distribution. It is easy to check that this condition characterizes the class of automodel tableaux. This fact can be regarded as "Kepler's second law for the transverse growth of trees". Indeed, assume that the graph of a diagram w describes the profile of a cut of a tree trunk. The shape of this profile will be stable up to similarity if and only if the intensity of appearance of new cells within an arc is proportional to the area of the sector supported by this arc. 5.6. The asymptotics of the general solution. We have shown in Section 5.4 that the diagram R is, up to similarity, the only fixed point of the basic dynamic equation (4.5.8). Let us now check that this solution is stable (attracting all other solutions). More precisely, each solution w(u, t) of (4.5.8) (with arbitrary initial conditions), after the normalization of its area, converges uniformly to R(u) ast+m.

55. DIFFERENTIAL MODEL O F GROWTH O F YOUNG DIAGRAMS

175

THEOREM.Assume that the charge a ( u , t) of a tableau w(u, t) satisfies (4.5.10). Then

unzformly in u E R. PROOF. The moments 5, (t) of the normalized diagram G(u, t ) = w(u&, t ) / & are easy to express in terms of the moments of the diagram w(u, t):

p, ( t )= t Let us find their asymptotics as t

-+

(t).

m , using (4.5.11).

LEMMA.If the functions p,(t), n

=

1 , 2 , .. . , satisfy

then there exist constants {P,),",~ such that

PROOF. The first equations of the system (4.5.20) are p;/3 = 0, pk/4 = 3(p? + ~2112,

+ pk/6 = 5 ( ~ + ? 6 ~ ?+ ~ 32 ~ + ; 8 ~ 1+ ~ 63~ 4 ) / 2 4 . Pk/5 = 4(P? + 3 ~ 1 ~ 22~ 3 ) / 6 , Successively solving them, we find that

where the c k are arbitrary constants. Since the right-hand side of (4.5.20) is homogeneous, the lemma follows easily by induction. To complete the proof of the theorem, it remains to check that the limiting values p, in (4.5.21) coincide with the moments of the diagram a. This can be done most conveniently by going over to equation (4.5.12). h, ( t ) ~ - ( ~ +is' )constructed from the normalized If the function R(X,t) = diagram G(u, t) = w(u&, t)/&, then

zr=o

176

4.

Y O U N G DIAGRAMS IN P R O B L E M S O F ANALYSIS

Substituting this expression into (4.5.12) yields, after simple calculations, the equat ion

Let us write the function R in the form

+

(4.5.23)

~ ( xt), = ~ ( x ) ~ ( xt ), ,

where ~ ( z =) C,"==, h , ~ - ( ~ + ' ) the , coefficients h, being related to pn in the usual way. Then, by the lemma, lim ~ ( xt), = 0,

t--*a

lim t ~ k ( xt), = 0.

t-m

Substituting (4.5.23) into (4.5.22) and letting t -+ m , we see that ZR - R2 = const, and the proof can be completed like those of the theorems of Sections 5.4 and 5.5 by using the quadratic equation R2(x) - z ~ ( z ) 1 = 0 and the formula for the generating function of the transition measure of R.

+

$6. Plancherel growth and semicircle diffusion

The transition distribution of the arcsine law R given by (4.5.1) is the semicircle distribution with density J-12~. In this book the arcsine law has appeared repeatedly in various contexts. The semicircle law also appears in many situations, of which we will consider only two: Wigner's semicircle law for the spectra of random matrices, and Voiculescu's free probability theory. In this section we consider the relation between the arcsine law and the semicircle law in more detail. 6.1. Wigner's semicircle law. Another context where the curve R appears is related to the spectral theory of random matrices and Wigner's semicircle law. Let A(") be a symmetric matrix of order n with characteristic polynomial

Denote by p(,) the discrete distribution with atoms of equal weights p p ) = l / n at the characteristic roots xk, k = 1,.. . , n. Denote by

the distribution function of the measure p("), and by F ( x ) the distribution function of the measure dp(u) = ( 2 ~ ) ~ ' du on the interval [-2,2]. We are interested in the case when the matrix A(,) is random; in this case the polynomial P,, its roots, the measure p(,), and the function F, are also random.

THEOREM (see [SS]). Let the matrix entries a/;) for i = 0 and ~(a!;)) = 1. Then, for every r > 0,

<j

be i.i.d. with ~(a:;))

56. PLANCHEREL GROWTH AND SEMICIRCLE DIFFUSION

This result can be restated in terms of diagrams. As above, let X I , . . . , x, denote the characteristic roots of the matrix ~ ( ~ and 1 , let y l , . . . , y,-1 be the extrema of the characteristic polynomial Pn(x) of A("); these two sequences interlace. Let 3, be the rectangle diagram constructed from the pair of sequences 2 1 < y1 < . . . < yn-1 < x,.

PROPOSITION.Under the assumptions of the theorem,

a s n - + m for every6 > 0 . The equivalence of this result and Wigner's theorem is a consequence of the remarkable Krein correspondence between diagrams and probability measures. Indeed, since

the diagram an constructed from the zeros and extrema of the polynomial corresponds to the uniform distribution on its roots, and (4.6.2) is equivalent to (4.6.1).

6.2. Small oscillations and the rigidity diagram. Consider a stable mechanical system with n degrees of freedom which oscillates near the equilibrium. Its potential energy is quadratic and determined by a symmetric positive definite matrix A. The squares of the fundamental frequencies of the system coincide with the eigenvalues X I , .. . , x, of this matrix. If a linear constraint is imposed on the system, the new fundamental frequencies will interlace with the original ones, where y l , . . . , y,-1 are the eigenvalues of the matrix A restricted to the corresponding hyperplane. The rectangular diagram wn describing (see Section 1.1) the separation of the squared frequencies of the linear system (4.6.4) under a linear constraint is called the rigidity diagram. Suppose now that the system (i.e., the matrix A) and the linear constraint (the hyperplane h c Rn) are chosen at random. It turns out that under broad conditions this random system is self-amenable (see [ 5 5 ] ) :as the number of degrees of freedom grows, the random rigidity diagram approaches (in an appropriate scaling) a nonrandom diagram. Let us show that this limiting diagram is exactly 0.

THEOREM Let. w, be the rigidity diagram of a random symmetric matrix A ( ~ ) of order r? with respect to a random hyperplane h c Rn. Assume that the normal to h is uniformly distributed o n the unit sphere i n Rn, and the matrix entries a (i y ) , i < j , are independent of h and among themselves, and are identically distributed ) 1. T h e n with mean value ~ ( a j ; ) )= 0 and variance ~ ( a ! ; ) =

uniformly i n u E R.

178

4.

PROOF.

YOUNG DIAGRAMS IN PROBLEMS OF ANALYSIS

Let us first average with respect t o the random constraint h .

L E M M A Let . Pn(x) = n ( x - xi) be the characteristic polynomial of a matrix A of order n, let h c Rn be a random hyperplane whose normal is uniformly distributed o n the unit sphere, and let Qn(x) = n ( x - yi) be the characteristic polynomial of the matrix Ah = pAp (where p is the orthogonal projection o n h ) . Then

PROOF.

Consider the partial fraction expansion of Qn/ Pn:

It is not difficult to see that pk = ti, where J = (I1,. . . , t n ) is the normal to h. Since the distribution of J is homogeneous, we have

because

C pk = 1. It remains to apply (4.6.3).

The theorem now follows from the semicircle law, stated as in (4.6.2). 6.3. The free convolution of probability distributions. In a series of papers and the monograph [167], D. Voiculescu constructed a deep analogue of probability theory which still has no generally accepted name. We will call it free probability the0ry.l It is based on an analogue of the ordinary convolution of two distributions p, v, which is called the free convolution [167] and is denoted by pmv. The problem leading to the definition of the free convolution can be stated as follows. Let G be the free group with two generators, and let Ul, U2 be the unitary operators in 12(G) corresponding to the left shifts by these generators. Denote by 5 = h9,, the vector from 12(G)determined by the 6-function supported by the unity of the group. Consider functions X I = cpl(Ul), X2 = cp2(U2)in the operators Ul, U2, and let p1, p2 be their spectral measures, i.e.,

for each polynomial f . As Voiculescu observed, the spectral measure of the sum X I + X 2 depends only on the spectral measures of the summands. This distribution p = p l El p2 is called the free convolution of the distributions p l , p2. Voiculescu [166] gave a purely analytical interpretation of the free convolution. The key notion here is the so-called R-transform. Following [166], denote by

the Cauchy-Stieltjes transform of a measure p which is a generating function of the moments m l , m2, . . . of this measure:

l Translation

Editor's Note. This name is now in general use

56. PLANCHEREL GROWTH AND SEMICIRCLE DIFFUSION

Then there exists a series

such that

The series (4.6.9) is called the R-transform of the measure p . With respect to the free convolution, it plays the role of the logarithm of the Fourier transform. Indeed, Voiculescu [I661 showed that the R-transform is additive with respect to the free convolution of measures:

Another useful fact from [I661 is that the identity (4.6.11) can be taken as an implicit definition of the free convolution. By analogy with ordinary probability theory, the coefficients rl ,r2,. . . of (4.6.9) could be called free semi-invariants. The first three of them coincide with the classical semi-invariants. The difference appears first for the coefficient r4, which differs from the classical semi-invariant

+

by the squared variance: r4 = s4 (m2 - m:)2. An essential difference of the free convolution from the ordinary one is its nonlinearity: the distributive law does not hold for B. Of course, the commutative and associative laws remain valid. An outstanding role in free probability theory is played by the semicircle distributions 1 (4.6.12) dpt(x) = -J a d x , 1x1 5 2t. 27rt It is easy to verify that the R-transform of these distributions is

It follows that

so that the distributions (4.6.12) form a semigroup with respect to the free convolution. Let us define the semicircle diffusion as the flow, in the space of measures M, determined by the formula

Voiculescu showed in the same paper [I661 that the Cauchy-Stieltjes transform

of the diffusing measure ut satisfies the Burgers equation

180

4.

YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS

Since the transform (4.6.7) uniquely determines the measure p , we may say that the Burgers equation provides an adequate analytical description of the semicircle diffusion. 6.4. The semicircle diffusion and the Plancherel growth of Young diagrams. Consider a rectangular diagram w E Do with extrema a t points

and define its small perturbation wt, t > 0. By definition, the diagram wt E Do is obtained from w by attaching a small square of area vkt, where vk are the Plancherel transition probabilities

above each minimum xk. Thus (see Figure 13 on p. 54) the rectangular diagram wt has minima at the points (4.6.18)

X:

= xk

d~

a,

and maxima at the points yk, xk. Consider the moment generating function of the diagram wt. By the general formula (4.1.14), this rational fraction equals

PROPOSITION. The function R(x, t ) defined by (4.6.19) satisfies the identity

PROOF.Let us rewrite (4.6.19) in the following form:

Since

we obtain dR(x, 0) R(x, t) - R(x, 0) = -tR(x, 0)-dx and (4.6.20) follows in the limit as t -+ 0.

+ o(t),

Thus we have checked that if a rectangular diagram wt grows according to the Plancherel probabilities (4.6.17), then its transition distribution vt infinitesimally (for small t ) evolves according t o formula (4.6.14), i.e., diffuses by means of the free convolution with the semigroup (4.6.12). The link is provided by the Burgers equation. Let us consider more carefully the behaviour of the transition measure vt of the rectangular diagram wt. We will explicitly describe a discrete approximation fit

$6. PLANCHEREL GROWTH AND SEMICIRCLE DIFFUSION

181

of the measure ut, and show that these two measures coincide up to terms of order o ( t ) as t -+ 0. By definition, the measure fit attaches the weights

to the points x:

= xk

f

m,where

Ak=C--ui

,

k = I , . . . ,n.

iZk X k - X j

PROPOSITION. Denote by

the Cauchy-Stieltjes transfomn of the measure fit (recall that the corresponding transform of the transition measure ut of the diagram wt is given b y (4.6.19)). Then

PROOF. Let us rewrite the general summand in (4.6.23) as follows: -

X - X Z +-X - x i Uk

Ak

-

x

-

Xk

+

( X - x k ) Vk A k uk t (X -~ k - ~) k ~t

(G+ ( X -ukX I ) '

)t+

O(t))

It follows that

where

We have earlier seen that the function R ( x , t ) satisfies the analogous identity

where

(4.6.28) In order to verify that C = C , subtract from both sides the second summand of (4.6.26); we arrive at the formula

182

4.

YOUNG DIAGRAMS IN PROBLEMS OF ANALYSIS

It remains to prove (4.6.29). Substituting the values of Ak from (4.6.22), we can rewrite (4.6.29) in the form

Since --v j

xk

-

'/j Xj

-

( X - x k ) vj (xk- x j )( X -x j )

X -X j

(4.6.30) yields

Using the identity

we can finally rewrite (4.6.31) in the form

But in this form the formula is obvious, and the proof is complete. Formula (4.6.21) illustrates the character of mass transfer of the original discrete distribution v in the course of the semicircle diffusion during a small period of time t. It is worth mentioning that the semicircle diffusion has a number of differences from the ordinary Gaussian diffusion: ( 1 ) Each mass vk breaks into two almost equal point masses v:. ( 2 ) The latter masses disperse as xk f i.e., heavy particles disperse faster. ( 3 ) The barycentre ( v z x t v i x i ) / ( v z v i ) of the dispersing masses v f is of order Akt, and the moment is equal t o v k t ~ ( t ) . ( 4 ) The particles do not go to infinity within a finite time, i.e., the measures v t , fit are compactly supported.

+

a, +

+

6.5. The diffusion equation in Voiculescu's theory. Let us give a heuristic argument showing that the equation

can be regarded as the diffusion equation in free probability theory. Unfortunately, this equation looks more complicated than the ordinary one, because of the nonlinear character of free convolution. Let us compute the mean value of a test function f with respect to the discrete measure fi defined by (4.6.21) and (4.6.22). Since

57. ASYMPTOTIC SEPARATION OF ROOTS OF ORTHOGONAL POLYNOMIALS

we see that for t

=0

183

the derivative of the mean value equals

and we arrive at an integral sum for the integral in the right-hand side of (4.6.32). 57. Asymptotic separation of roots of orthogonal polynomials The main purpose of this section is to show that a wide class of orthogonal polynomials (including the classical polynomials of Jacobi, Hermite, and Laguerre) has a common universal asymptotics of mutual separation of roots. We emphasize that the question here is not in the distribution of roots, but in the character of their mutual separation. The section is organized as follows: we state the main result in Sections 7.1-7.2, and prove it in Sections 7.3-7.4. Section 7.5 contains some estimates which refine the main theorem in the case of the Chebyshev polynon~ials.

7.1. Separation of roots of orthogonal polynomials. Consider polynomials n

belonging to a family of polynomials orthogonal with respect to a certain measure dp(x). It is well-known ( [ 5 6 ] )that their roots satisfy the separation condition (4.1.2). For example, Table 1 presents the roots (for n = 15,16) of the Chebyshev polynomials of the second kind Un(x) =

+

sin (n 1) arccos x sin arccos x

1

the Hermite polynomials

and the Laguerre polynomials

Figure 11 (on p. 47) displays the corresponding rectangular diagrams. We see that the diagrams of root separation in this figure are strikingly alike (up t o scale parameters). To emphasize this fact, we have laid the (appropriately scaled) graph of the function

upon these diagrams. Note that the area A v < Q(u)) of this function is equal to one.

=

A(R) of the subgraph {(u, v): lul 5

The function R appeared first in [12]and [I381 in connection with a problem of the asymptotic representation theory of the symmetric group (see 52 in Chapter 3 for the description of the limit shape of large Young diagrams). We will show that

184

4.

YOUNG DIAGRAMS IN PROBLEMS OF ANALYSIS

-

-

-

TABLE1 The roots of polynomials (of degrees 16 and 15) Hermite Laguerre Chebyshev

this function describes the asymptotics of the root separation for a wide class of orthogonal polynomials. 7.2. Linear recurrence relations. I11 what follows it will be more convenient t o use another standard approach to the theory of orthogonal polynomials when they arise as solutions of difference equations (see, e.g., [2]). Consider a second order linear difference equation

with spectral parameter u, and let 4, = P n ( u ) , n = 0,1,2, . . . , be the unique solution of this equation satisfying the initial conditions

Obviously, Pn(u) is a monic polynomial of degree n. Assuming that c i > 0 for all n 2 1, one can easily see that the roots of the polynomials Pn(u) are real and simple, and the roots of two adjacent polynomials Pn-l(u), Pn(u) interlace (see [2], Chapter 4). Favard's theorem (see [52], Chapter 8.6) guarantees that there exists a measure p for which the polynomials Pn are orthogonal: 00

(4.7.4)

P,(u) Pn(u)dp(u) = 0 f o r m

# n.

Conversely, the polynomials orthogonal with respect to a certain measure satisfy a linear recurrence relation of the form (4.7.2). Consider the family of orthogonal polynomials { P , ( U ) ) ~ =determined ~ by the linear recurrence relation

57. ASYMPTOTIC SEPARATION O F ROOTS O F ORTHOGONAL POLYNOMIALS

185

It is well-known that the roots of the adjacent polynomials

interlace: x1 < yl < x2 < . . . < xn-1 < yn-1 < x,. Let us associate with the pair of polynomials (4.7.6) a continuous piecewise linear function v = w n ( u ) (we call it a diagram) such that

Pn-1 ( u ) wk ( u ) = sign pn(u) ' w n ( u ) = Ju- znI for sufficiently large ( u ( , where zn = C x i - C yi. The roots of Pn are the minima of w,, and the roots of Pn-l, its maxima (Figure 21, on p. 145). The key result of this section is the following theorem.

THEOREM Let. P n ( u ) be the orthogonal polynomials determined by equation (4.7.2) with initial conditions (4.7.3), and let wn be the rectangular diagram describing the mutual separation of roots of the polynomials P n - l ( u ) , P n ( u ) . Assume that the coeficients of (4.7.2) satisfy the following conditions: Cn-1

lim -= 1, cn

(4.7.8)

n+m

bn+l - bn cn

lim

n-m

= 0.

Then lim

n-cc

1 cn

-W,(UC~-~

+ bn) = O ( u )

~lniformlyi n u E R, where the function R is defined by (4.7.1). In [I181this result was established with less generality: instead of conditions (4.7.8) it was assumed that

(4.7.10)

lim c n = c > O ,

n-00

lim bn=O.

n-cc

Let d p be the measure for which the family {P,) is orthogonal. For condition (4.7.10) to be satisfied, it is sufficient that d p ( u ) be supported by the interval [ - I l l ] and the function log pf(cos0) be integrable on the interval [0,T]. On the other hand, the theorem certainly fails if the support of d p is not connected. The classical Hermite and Laguerre polynomials are in an intermediate position: they satisfy (4.7.8), but not (4.7.10). It is known [56]that if c = 1 in (4.7.10), then there exists a common limiting density of the distribution of roots: lim

n+cc

jul

n.

I 1.

The roots of Hermite polynomials have a different limiting density:

1 Xi lim -#{i: - < U ) = n-cc n

6

2T

-1

JTZ?dxl

1111

< 11

but the asymptotics of mutual separation of roots is the same in both cases. Thus our result is more universal (but also coarser) than the theorems on the limiting distribution of roots of orthogonal polynomials.

186

4.

YOUNG DIAGRAMS IN PROBLEMS O F ANALYSIS

7.3. T h e e q u a t i o n for t h e limiting R-function. Let Rn (u) = Pn- 1(u)/Pn (u) be the R-function of the diagram w,. Dividing both sides of the identity Pn+l(u) = ( U

-

bn+l)Pn(u) - c i ~ n - l ( u )

by Pn(u), we obtain a recurrence relation for the R-functions:

Using the formulas from Section 2.5, we go over to the normalized functions, i.e., the R-functions

+

of the diagrams Gn(u) = wn(ucn-l bn) / cn-1, Gn+i (u) = wn+l(ucn + bn+l) / cn. After we replace u by (u - bn+l)/cn, formula (4.7.11) takes the form

where A, = bn+l - b,. Assume that the limit

exists for sufficiently large 1ul. Then in view of (4.7.8) we obtain the equation

from which it follows that R ( U ) = i ( u - d m ) = Rn(u) (the sign before the square root is chosen from the condition ~ ( m=) lim ~ , ( m )= 0). In order to prove that the limit (4.7.12) exists, we will consider the behaviour of the Taylor coefficients at infinity.

7.4. T h e Taylor coefficients. Let us rewrite (4.7.11) in the form

and consider the power series

where

57. ASYMPTOTIC SEPARATION OF ROOTS OF ORTHOGONAL POLYNOMIALS

187

we obtain recurrence relations for the coefficients:

for k

= 0,1,2,

More generally, (4.7.14) and (4.7.15) imply LEMMA.The coefficients r P + ' ) are homogeneous polynomials of degree k in the first k - 1 variables of the sequence c,, A,, c,-1, An-l,. . . . COROLLARY.Under the assumptions of the theorem, the limits lim,,, fP)= Ck exist, where f r ) = r?)/ck are the Taylor coefficients of the normalized R function

&

(2kk) are the Catalan numbers. Note that the limiting coefficients C2k = These numbers also arise as the coefficients of the expansion Rn(u) = Co/u C2/u3 + . . . of the R-function of the diagram R. The convergence of R-functions lim R, (u)= Rn (u) implies the uniform convergence of diagrams lim Gn(u) = R (u), and the theorem follows.

+

7.5. Separation of roots of the Chebyshev polynomials. In this section we give a direct proof of Theorem 7.2 in the special case of the Chebyshev polynomials of the second kind. The proof contains a simple bound on the rate of convergence in (4.7.9).

s,

PROPOSITION.Let xk = - cos k = 1,.. . , n , be the roots of the Chebyshev polynomials of the second kind U, (x), and let yk = - cos $, k = 1, . . . , n- 1, be the roots of Un-l(x). Denote by w, the rectangular diagram describing the interlacing sequences {xk), {yk). Then (4.7.16) as n

I ~ n ( u) R(u)I

--, co,uniformly

= O(l/n)

in u E R.

PROOF. It suffices t o prove (4.7.16) for u the definition (4.7.1) that

= xk

and for u

= yk.

It is clear from

By the description (4.1.4) of the diagram w, we have k

w ( Y ~=) 2 ~ k- (yk - Z )

=

-yk

+ 2~~,

where zk

=x

( y i - xi).

i= 1

L

U!S

"41+ YZ)U!S

-

T+"D

U!S

+

= YZZ

TfUo(l ~Z)U!S

q n u n o j ay? Lq sau!sor,

wyq aas am

JO

urns ayl 8u!Ljgdur!s

References 1. V. I. Arnold, Mathematical Methods of Classical Mechanics, "Nauka", Moscow, 1974; English transl., Springer-Verlag, Berlin, 1978. 2. F. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York, 1964. 3. M. Atiyah, K-theory, Benjamin, New York, 1967. 4. M. I. Akhiezer, The Classical Moment Problem and Some Related Questions i n Analysis, Fizmatgiz, Moscow, 1961; English transl., Hafner, New York, 1965. 5. A. Erdklyi et al., Higher Transcendental Functions. Vol. 11, McGraw-Hill, New York, 1953; reprint, Krieger, Melbourne, FL, 1981. 6. M. Sh. Birman and D. R. Yafaev, The spectral shift function. T h e papers of M. G . Krein and their further development, Algebra i Analiz 4 (1992), no. 5, 1-44; English transl., St. Petersburg Math. J . 4 (1993), 833-870. 7. B. L. van der Waerden, Algebra, Ungar, New York, 1970 (among many other editions in several languages). 8. H. Weyl, The Classical Groups. Their Invanants and Representations, Princeton Univ. Press, Princeton, NJ, 1939. 9. A. M. Vershik and A. A. Shmidt, Symmetric groups of higher degree, Dokl. Akad. Nauk SSSR 206 (1972), 269-272; English transl., Soviet Math. Dokl. 13 (1972), 1190-1194. 10. A. M. Vershik, Description of invariant measures for the actions of some infinite-dimensional groups, Dokl. Akad. Nauk SSSR 218 (1974), 749-752; English transl., Soviet Math. Dokl. 15 (1974), 1396-1400. 11. , The hook f o m u l a and related identities, Zapiski Nauchn. Semin. LOMI 172 (1989), 3-20; English transl., J. Soviet Math. 59 (1992), 1029-1040. 12. A. M. Vershik and S. V. Kerov, Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tableaux, Dokl. Akad. Nauk SSSR 233 (1977), 10241027; English transl., Soviet Math. Dokl. 18 (1977), 527-531. 13. , Asymptotic theory of characters of the symmetric group, Funkts. Anal. i Prilozhen. 15 (1981), no. 4, 15-27; English transl., Funct. Anal. Appl. 15 (1982), 246-255. 14. , Characters and factor representations of the infinite symmetric group, Dokl. Akad. Nauk SSSR 257 (1981), 1037-1040; English transl., Soviet Math. Dokl. 23 (1981), 389-392. 15. , Characters and factor representations of the inJnite unitary group, Dokl. Akad. Nauk SSSR 267 (1982), 272-276; English transl., Soviet Math. Dokl. 26 (1982), 570-574. 16. , The K-functor (Grothendieck group) of the inJnite symmetric group, Zapiski Nauchn. Semin. LOMI 123 (1983), 126-151; English transl., J . Soviet Math. 28 (1985), 549-568. 17. , Locally semisimple algebras. Combinatorial theory and the K-functor, Itogi Nauki i Tekhniki, Ser. Sovrem. Probl. Mat., vol. 26, VINITI, Moscow, 1985, pp. 3-56; English transl., J. Soviet Math. 38 (1987), 1701-1733. 18. , Asymptotic behavior of the m a x i m u m and generic dimensions of irreducible representations of the symmetric group, Funkts. Anal. i Prilozhen. 19 (1985), no. 1, 25-36; English transl., Funct. Anal. Appl. 19 (1985), 21-31. Characters and realizations of representations of the infinite-dimensional Hecke 19. , algebra, and knot invariants, Dokl. Akad. Nauk SSSR 301 (1988), 777-780; English transl., Soviet Math. Dokl. 38 (1989), 134-137. 20. A. M. Vershik and A. A. Shmidt, Limit measures arising i n the asymptotic theory of symmetric groups. I, Teor. Veroyatnost. i Primenen. 22 (1977), 72-88; English transl., Theory Probab. Appl. 22 (1977), 70-85.

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Comments on Kerov's Thesis G. OLSHANSKI Chapter 1 1) Section 1, The ergodic method. In connection with the ergodic theorem proved in the text, see also Olshanski and Vershik [A.46] and Olshanski [A.44], Proposition 10.8. 2) Section 3, Theorem 1 (Boundary of Kingman's graph). Both Kingman's theorem and Thoma's theorem are particular cases of a more general result proved by Kerov, Okounkov, and Olshanski [A.34].

Chapter 2 3) Section 2.2, Asymptotics of the number of skew Young diagrams. Detailed proofs and further results- are contained in Wassermann [A.55], in Okounkov and Olshanski [A.41], and in Olshanski, Regev, and Vershik [A.45].

4) Section 2.3, Thoma's theorem. A generalization of Thoma's theorem is given by Kerov, Okounkov, and 01shanski [A.34]. The proof is based on Vershik and Kerov's asymptotic method, but does not use the multiplicativity property.

5) Section 9.3, The conjecture. One more case when the Kerov conjecture is known to be true is that of Jack polynomials, which are a degenerate case of Macdonald's polynomials Px(x; q, t ) (t = q1Ia, q -4 1, a > 0 is fixed). See Kerov, Okounkov, and Olshanski [A.34]. In the general case, the conjecture remains open, and any advance in this direction would be very interesting. In particular, the case of Hall-Littlewood polynomials (q = 0) is interesting because of connections with the results of Kerov and Vershik [A.54] on the characters of certain infinite matrix groups over finite fields. Chapter 3 6) Section 1.4, The Ulam problem. A new achievement related to the Ulam problem is due t o Baik, Deift, and Johansson [A.3]: they found the asymptotics of the distribution of 7-1(A) (the length of the first row of the random Plancherel diagram). It turned out that the result coincides with the asymptotic distribution for the largest eigenvalue of the random Hermitian N x N matrix with Gaussian measure, N -+ co. Further results in this direction: Baik, Deift, and Johansson [A.4], Okounkov [A.37], Borodin,

198

COMMENTS ON KEROV'S THESIS

Okounkov, and Olshanski [A.11], and Johansson [A.30]; see also the expository paper by Deift [A.15].

7) Section 2, Gaussian limit of the Plancherel measure. In the late 90s, Kerov returned to this subject and found another approach to the results stated in Section 2.1. His proof was reconstructed by Ivanov and 01shanski [A.28]. As shown in that paper, Kerov's method also allows one to describe the fluctuations of the transition measures of the random Plancherel diagrams; the result shows a striking similarity with the central limit theorem for random matrices (Diaconis and Shahshahani [A.161, and Johansson [A.29]). Kerov's new approach uses the results of his joint paper with Ivanov [A.27], which contains a detailed proof of Proposition 1 and further results about convolution of conjugacy classes in symmetric groups. The algebra of functions on Young diagrams A, briefly mentioned in the very beginning of Section 2.3, is an important object; in more detail it is considered in Kerov and Olshanski [124], see also Lascoux and Thibon [A.36]. This algebra (which is denoted in [A.28] as A) can be identified with the algebra of shifted symmetric functions in the row coordinates of X (see Okounkov and Olshanski [A.41]). Using the modified Frobenius coordinates of A, the same algebra A can also be identified with the algebra of symmetric functions in its "super" realization (see Olshanski, Regev, and Vershik [A.45]). Kerov's approach is largely based on exploiting relations between various natural bases and systems of generators in the algebra A. Here there are a number of open questions, see, e.g., Biane [A.7], [A.8]. An alternative proof of Theorem 1 is given by Hora [A.25]. An analog of Theorem 1 for characters of projective representations of symmetric groups was obtained by Ivanov [A.26]; his paper incorporates a "projective" version of the results of [A.27].

8) Section 3, Distribution of symmetry types for tensors of large degree. , ~ regime, when Biane studied the asymptotics of the measure p ~ in another m const N ' / ~ Then . a different limit shape arises, see [A.6], [A.7], [A.8]. 9) Theorem 2 (Section 3.4) was rediscovered by Tracy and Widom [A.50] and by Johansson [A.30].

Chapter 4 10) Kerov's ideas related to the concept of continual diagrams and their transition measures were further developed and exploited by Biane, see [A.5], [A.9]. 11) The material of Sections 1-3 is discussed in more detail and greater generality in Kerov's paper [A.33]. Further results concerning the hook walk algorithm of Section 4 were obtained by Romik [A.47].

Additional References A.1. D. Aldous and P. Diaconis, Longest increasing subsequences: from patience sorting to the Baik-Deifl-Johansson theorem, Bull. Amer. Math. Soc. (N.S.) 36 (1999), 413-432. A.2. M. Adler and P. van Moerbeke, Integrals over classical groups, random permutations, Toda and Toeplitz lattices, Comm. Pure Appl. Math. 54 (2001), 153-205. A.3. J. Baik, P. Deift, and K. Johansson, O n the distribution of the length of the longest increasing subsequence of random permutations, J . Amer. Math. Soc. 12 (1999), 1119-1178. O n the distribution of the length of the second row of a Young diagram under A.4. , Plancherel measure, Geom. Funct. Anal. 10 (2000), 702-731. A.5. P. Biane, Representations of symmetric groups and free probability, Adv. Math. 138 (1998), 126-181. A.6. , Approximate factorization and concentration for characters of symmetric groups, Internat. Math. Res. Notices 2001, 179-192. A.7. , Free cumulants and representations of large symmetric groups, XIIIth Internat. Congr. Math. Phys. (London, 2000), International Press, Boston, MA, 2001, pp. 321-326. A.8. , Free cumulants and characters of symmetric groups, Preprint, 2001. Free probability and combinatorics, Proc. Internat. Congr. Math. (Beijing, 2002), A.9. , Vol. 11, World Sci. Publ., Singapore, 2002. Characters of symmetric groups and free cumulants, in [A.56],pp. 185-200. A.lO. , A . l l . A. Borodin, A. Okounkov, and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups, J . Amer. Math. Soc. 13 (2000), 481-515. A.12. A. Borodin and G. Olshanski, Infinite random matrices and ergodic measures, Comm. Math. Phys. 223 (2001), 87-123. z-measures o n partitions, Robinson-Schensted-Knuth correspondence, and P = 2 A.13. , random matrix ensembles, Random Matrix Models and Their Applications, Math. Sci. Res. Inst. Publ., vol. 40, Cambridge Univ. Press, Cambridge, 2001, pp. 71-94. A.14. M. Bozejko, B. Kiimmerer, and R. Speicher, q-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys. 185 (1997), 129-154. A.15. P. Deift, Integrable systems and combinatorial theory, Notices Amer. Math. Soc. 47 (2000), 631-640. A.16. P. Diaconis and M. Shahshahani, O n the eigenvalues of random matrices, J. Appl. Probab. 31A (1994), 49-62. A.17. J. Fulman, A probabilistic approach toward conjugacy classes i n the finite general linear group, J . Algebra 212 (1999), 557-590. A.18. , New examples of potential theory o n Bratteli diagrams, Preprint, 1999, arXiv: math.C0/9912148. A.19. A. Gnedin, The representation of composition structures, Ann. Probab. 25 (1997), 14371450. Three sampling formulas, Preprint, 2002, arXiv:math.PR/0210319. A.20. , A.21. A. Gnedin and S. Kerov, The Plancherel measure of the Young-Fibonacci graph, Math. Proc. Cambridge Philos. Soc. 129 (2000), 433-446. A.22. , A characterization of GEM distributions, Combin. Probab. Comput. 10 (2001), 213-217. A.23. , Fibonacci solitaire, Random Structures Algorithms 20 (2002), 71-88. A.24. F. Goodman and S. Kerov, The Martin boundary of the Young-Fibonacci lattice, J . Algebraic Corrtbin. 11 (2000), 17-48. A.25. A. Hora, Central limit theorem for the adjacency operators o n the infinite symmetric group, Comm. Math. Phys. 195 (1998), 4055416.

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A.26. V . N. Ivanov, Gaussian limit for projective characters of large symmetric groups, Zapiski Nauchn. Semin. POMI 283 (2001),73-97; English transl., t o appear in J . Math. Sci. (New York). A.27. V . Ivanov and S. Kerov, The algebra of conjugacy classes i n symmetric groups, and partial permutations, Zapiski Nauchn. Semin. POMI 256 (1999),95-120; English transl., J . Math. Sci. (New Y o r k ) 107 (2001),4212-4230. A.28. V . Ivanov and G . Olshanski, Kerov's central limit theorem for the Plancherel measure on Young diagrams, Symmetric Functions 2001: Surveys o f Developments and Perspectives ( S . Fomin, ed.), Kluwer, Dordrecht, 2002, pp. 93-151. A.29. K . Johansson, O n jluctuations of eigenvalues of random Hermatian matrices, Duke Math. J. 91 (1998), 151-204. A.30. , Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. o f Math. ( 2 ) 153 (2001),259-296. A.31. S. Kerov, Double function algebras on a finite group, Zap. Nauchn. Sem. LOMI 39 (1974), 182-185; English transl., J . Soviet Math. 8 (1977), 136-139. A.32. , Duality of finite-dimensional *-algebras, Vestnik Leningrad. Univ. 1974,no. 7 (Ser. Mat. Mekh. A s h . vyp. 2 ) , 23-29; English transl., Vestnik Leningrad Univ. Math. 7 (1979), 122-130. A.33. , Interlacing measures, Kirillov's Seminar on Representation Theory, Amer. Math. Soc. Transl. ( 2 ) 181 (1998),35-83. A.34. S. Kerov, A . Okounkov, and G . Olshanski, The boundary of Young graph with Jack edge multiplicities, Internat. Math. Res. Notices 1998,173-199. A.35. S. V . Kerov and N. V . Tsilevich, The Markov-Krein correspondence i n several dimensions, Zapiski Nauchn. Semin. POMI 283 (2001),98-122; English transl., t o appear in J . Math. Sci. (New Y o r k ) . A.36. A . Lascoux and J.-Y. Thibon, Vertex operators and the class algebras of the symmetric groups, Zapiski Nauchn. Semin. POMI 283 (2001), 156-177; English transl., t o appear in J . Math. Sci. ( N e w York).. A.37. A . Okounkov, Random matrices and random permutations, Internat. Math. Res. Notices 2000,,1043-1095. A.38. -, Infinite wedge and random partitions, Selecta Math. (N.S.) 7 (2001), 1-25. A.39. -, S L ( 2 ) and z-measures, Random Matrix Models and Their Applications, Math. Sci. Res. Inst. Publ., vol. 40, Cambridge Univ. Press, Cambridge, 2001, pp. 407-420. A.40. , Random trees and moduli of curves, in [A.56], pp. 89-126. A.41. A . Okounkov and G . Olshanski, Shijled Schur functions, Algebra i Analiz 9 (1997), no. 2, 73-146; English transl., St. Petersburg Math. J . 9 (1998),239-300. A.42. A . Okounkov and R . Pandharipande, Gromov- Witten theory, Hunuitz theory, and completed cycles, Preprint, 2002, arXiv :math.AG/0204305. A.43. , The equivariant Gromov-Witten theory of P1,Preprint, 2002, arXiv:math.AG/ 0207233. A.44. G.Olshanski, The problem of harmonic analysis on the infinite-dimensional unitary group, t o appear in J . Funct. Anal.; available via arXiv:math.RT/0109193. A.45. G . Olshanski, A . Regev, and A . Vershik, Frobenius-Schur functions, Studies in Memory o f Issai Schur ( A . Joseph et al., eds.), Progr. Math., vol. 210, Birkhauser, Basel, 2003, pp. 251-300. A.46. G . Olshanski and A . Vershik, Ergodic unitanly invariant measures on the space of infinite Hermitian matrices, Contemporary Mathematical Physics: F. A . Berezin Memorial Volume, Amer. Math. Soc. Transl. ( 2 ) 175 (1996), 137-175. A.47. D. Romik, Explicit formulas for hook walks on continual Young diagrams, Preprint. A.48. R . Speicher, Combinatorial theory of the free product with amalgamation and operatorvalued free probability theory, Mem. Amer. Math. Soc. 132 (1998),no. 627. pp. 53-74. A.49. -, Free probability theory and random matrices, in [A.56], A.50. C. A . Tracy and H . W i d o m , O n the distnbutions of the lengths of the longest monotone subsequences i n random words, Probab. Theory Related Fields 119 (2001),350-380. A.51. N . Tsilevich, Distribution of mean values of some random matrices, Zapiski Nauchn. Semin. POMI 240 (1997),268-279; English transl., J . Math. Sci. (New Y o r k ) 96 (1999),3616-3623.

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Titles in This Series S. V. Kerov, Asymptotic representation theory of the symmetric group and its applications to analysis, 2003 Kenji Ueno, Algebraic geometry 3: Further study of schemes, 2003 Masaki Kashiwara, D-modules and microlocal calculus, 2003 G. V. Badalyan, Quasipower series and quasianalytic classes of functions, 2002 Tatsuo Kimura, Introduction t o prehomogeneous vector spaces, 2003 L. S. Grinblat, Algebras of sets and combinatorics, 2002 V. N. Sachkov and V. E. Tarakanov, Combinatorics of nonnegative matrices, 2002 A. V. Mel'nikov, S. N. Volkov, and M. L. Nechaev, Mathematics of financial obligations, 2002 Takeo Ohsawa, Analysis of several complex variables, 2002 Toshitake Kohno, Conformal field theory and topology, 2002 Yasumasa Nishiura, Far-from-equilibrium dynamics, 2002 Yukio Matsumoto, An introduction to Morse theory, 2002 Ken'ichi Ohshika, Discrete groups, 2002 Yuji Shimizu and Kenji Ueno, Advances in moduli theory, 2002 Seiki Nishikawa, Variational problems in geometry, 2001 A. M. Vinogradov, Cohomological analysis of partial differential equations and Secondary Calculus, 2001 Te Sun Han and Kingo Kobayashi, Mathematics of information and coding, 2002 V. P. Maslov and G. A. Omel'yanov, Geometric asymptotics for nonlinear PDE. 1: 2001 Shigeyuki Morita, Geometry of differential forms, 2001 V. V. Prasolov and V. M. Tikhomirov, Geometry, 2001 Shigeyuki Morita, Geometry of charact.eristic classes, 2001 V. A. Smirnov, Simplicia1 and operad methods in algebraic topology, 2001 Kenji Ueno, Algebraic geometry 2: Sheaves and cohomology, 2001 Yu. N. Lin'kov, Asymptotic statistical methods for stochastic processes, 2001 Minoru Wakimoto, Infinite-dimensional Lie algebras, 2001 Valery B. Nevzorov, Records: Mathematical theory, 2001 Toshio Nishino, Function theory in several complex variables, 2001 Yu. P. Solovyov and E. V. Troitsky, CL-algebras and elliptic operators in differential topology, 2001 Shun-ichi Amari and Hiroshi Nagaoka, Methods of information geometry, 2000 Alexander N. Starkov, Dynamical systems on homogeneous spaces, 2000 Mitsuru Ikawa, Hyperbolic partial differential equations and wave phenomena, 2000 V. V. Buldygin and Yu. V. Kozachenko, Metric characterization of random variables and random processes, 2000 A. V. Fursikov, Optimal control of distributed systems. Theory and applications, 2000 Kazuya Kato, Nobushige Kurokawa, and Takeshi Saito, Number theory 1: Fermat's dream, 2000 Kenji Ueno, Algebraic Geometry 1: From algebraic varieties to schemes, 1999 A. V. Mel'nikov, Financial markets, 1999 Hajime Sato, Algebraic topology: an intuitive approach, 1999 I. S. Krasil'shchik and A. M. Vinogradov, Editors, Symmetries and conservation laws for differential equations of mathematical physics, 1999 Ya. G. Berkovich and E. M. Zhmud', Characters of finite groups. Part 2, 1999

TITLES IN THIS SERIES

A. A. Milyutin and N. P. Osmolovskii, Calculus of variations and optimal control, 1998 V. E. Voskresenskii, Algebraic groups and their birational invariants, 1998 Mitsuo Morimoto, Analytic functionals on the sphere, 1998 Satoru Igari, Real analysis-with an introduction to wavelet theory, 1998 L. M. Lerman and Ya. L. Umanskiy, Four-dimensional integrable Hamiltonian systems with simple singular points (topological aspects), 1998 S. K. Godunov, Modern aspects of linear algebra, 1998 Ya-Zhe Chen and Lan-Cheng Wu, Second order elliptic equations and elliptic systems, 1998 Yu. A. Davydov, M. A. Lifshits, and N. V. Smorodina, Local properties of distributions of stochastic functionals, 1998 Ya. G. Berkovich and E. M. Zhmud', Characters of finite groups. Part 1, 1998 E. M. Landis, Second order equations of elliptic and parabolic type, 1998 Viktor Prasolov and Yuri Solovyev, Elliptic functions and elliptic integrals, 1997 S. K. Godunov, Ordinary differential equations with constant coefficient, 1997 Junjiro Noguchi, Introduction to complex analysis, 1998 Masaya Yamaguti, Masayoshi Hata, and J u n Kigami, Mathematics of fractals, 1997 Kenji Ueno, An introduction to algebraic geometry, 1997 V. V. Ishkhanov, B. B. Lur'e, and D. K. Faddeev, The embedding problem in Galois theory, 1997 E. I. Gordon, Nonstandard methods in commutative harmonic analysis, 1997 A. Ya. Dorogovtsev, D. S. Silvestrov, A. V. Skorokhod, and M. I. Yadrenko, Probability theory: Collection of problems, 1997 M. V. Boldin, G. I. Simonova, and Yu. N. Tyurin, Sign-based methods in linear statistical models, 1997 Michael Blank, Discreteness and continuity in problems of chaotic dynamics, 1997 V. G. Osmolovskii, Linear and nonlinear perturbations of the operator div, 1997 S. Ya. Khavinson, Best approximation by linear superpositions (approximate nomography), 1997 Hideki Omori, Infinite-dimensional Lie groups, 1997 V. B. Kolmanovskii and L. E. Shaikhet, Control of systems with aftereffect, 1996 V. N. Shevchenko, Qualitative topics in integer linear programming, 1997 Yu. Safarov and D. Vassiliev, The asymptotic distribution of eigenvalues of partial differential operators, 1997 V. V. Prasolov and A. B. Sossinsky, Knots, links, braids and 3-manifolds. An introduction t o the new invariants in low-dimensional topology, 1997 S. Kh. Aranson, G. R. Belitsky, and E. V. Zhuzhoma, Introduction to the qualitative theory of dynamical systems on surfaces, 1996 R. S. Ismagilov, Representations of infinite-dimensional groups, 1996 S. Yu. Slavyanov, Asymptotic solutions of the one-dimensional Schrodinger equation, 1996 B. Ya. Levin, Lectures on entire functions, 1996 Takashi Sakai, Riemannian geometry, 1996

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/.

Asymptotic representation theory of symmetric groups deals with two types of problems: asymptotic properties of representations of symmetric groups of large order, and representations of the limiting object, i.e., the infinite symmetric group. The author contributed significantly in the development of problems of both types, and his book presents an account of these contributions, as well as those of other researchers. Among the problems of the first type, the author discusses the properties of the distribution of the normalized cycle length in a random permutation, and the limiting shape of a random (with respect to the Plancherel measure) Young diagram. He also studies stochastic properties of the deviations of random diagrams from the limiting curve. Among the problems of the second type, the author studies an important problem of computing irreducible characters of the infinite symmetric group. This leads him to the study of a continuous analog of the notion of Young diagram, and, in particular, to a continuous analogue of the hook walk algorithm, which is well known in the combinatorics of finite Young diagrams. In turn, this construction provides a completely new description of the relation between the classical moment problems of A Hausdorff and Markov.

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EDITORIAL COMMITTEE AMS Subcommittee Robert D. MacPherson Grigorii A. Margulis James D. Stasheff (Chair) ASL Subcommittee Steffen Lempp (Chair) IMS Subcommittee Mark I. Freidlin (Chair)

Tkanslated from t h e Russian manuscript by N. V. Tsilevich 2000 Mathematics Subject Classijication. P r i m a r y 20C30, 20P05, 22D10.

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Library of Congress Cataloging-in-Publication Data Kerov, S. V. (Sergei Vasil'evich), 1946-2000. [Asimptoticheskaia teoriia predstavleniia simmetricheskoi gruppy i ee primeneniia v analize. English] Asymptotic representation theory of the symmetric group and its applications in analysis / S. V. Kerov ; translated by N. Tsilevich. p. cm. - (Translations of mathematical monographs, ISSN 0065-9282 ; v. 219) Includes bibliographical references. ISBN 0-8218-3440-1 (acid-free paper) 1. Symmetry groups-Asymptotic theory. 2. Representations of groups. I. Title. 11. Series.

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