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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
1628
Springer
Berlin -Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
Paul-Hermann Zieschang
An Algebraic Approach to Association Schemes
~ Springer
Author Paul-Hermann Zieschang
Christian-Albrechts-Universifftt zu Kiel Mathematisches Seminar Ludewig-Meyn-StraBe 4 24098 Kiel, Germany
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Ziesehang, Paul-Hermann: An algebraic approach to association schemes / Paul-Hermann Zieschang. - Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Tokyo : S p r i n g e r , 1996 (Lecture notes in mathematics ; 1628) ISBN 3-540-61400-1 NE: GT Mathematics Subject Classification (1991): 05-02, 05E30, 51E12, 51E24, 05B25, 16S50 ISSN 0075-8434 ISBN 3-540-61400-1 Springer-Verlag Berlin Heidelberg New Y o r k
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.
9 Springer-Verlag Berlin Heidelberg 1996 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 1047964 46/3142-543210 - Printed on acid-free paper
Introduction
Let X be a set. We define
l x :-- {(x, x)
I e x).
Let r C_ X x X be given. We set
r" := {(y, z) I(z, Y) 9 r}, and, for each x 9 X, we define
xr := {y 9 x I (x, y) 9 r}. Let G be a partition of X • X such that 0 ~ G 9 1x, and assume that, for each g 9 G, g* 9 G. Then the pair (X, G) will be called an association scheme if, for all d, e, f 9 G, there exists a cardinal number age! such that, for all y, z 9 X,
(y, z) 9 f ~
lyd N ze* I = ade].l
In these notes, we shall always say scheme instead of association scheme. The pair (X, G) will always denote a scheme. We shall always write 1 instead of l x . The elements of {adeS I d, e, f 9 G} will be called the structure constants of (X, G).2 Occasionally we shall use the expression regularity condition in order to denote the condition which guarantees the existence of the structure constants. The present text provides an algebraic approach to schemes. Similar to the theory of groups, the theory of schemes will be viewed as an elementary 1 This definition of association schemes is slightly more general than the usual one. Usually one requires additionally at least that IX ] be finite. In this case, the term "homogeneous coherent configuration" is common, too; see [14]. (Also in the present text the finiteness of IX I plays an important role.) It is also often required that, for all d, e, f E G, aa~! = a~al; see, e.g., [1]. The term "association scheme" was introduced in [2]. There it is even required that, for each g E G, g* = g, and this additional condition (which implies that, for all d, e, f E G, age! = a~df ) is often included in the definition of association schemes. In order to emphasize the algebraic treatment, association schemes (in the present sense) were called "generalized groups" in [34]. 2 In the literature, the structure constants are also called "intersection numbers".
VI algebraic theory which is naturally connected to certain geometric structures. In fact, the theory of schemes generalizes naturally the theory of groups. Let us first see to what extent the class of groups m a y be viewed as a distinguished class of schemes. For each g E G, we abbreviate ng : : agg, 1.
A n o n - e m p t y subset F of G will be called thin if {1} = {n! I f E F } . (Note that we always have nl = 1.) T h e pair (X, G) will be called thin if G is thin. Let E, F C_ G be given. We define
EF:=lgeVl
:aojg r eEE f E F
and call it the complex product of E and F. It follows readily from the definition of the complex product that, if (X, G) is thin, G ( X , G ) := {{g} I g E G} is a group with respect to the complex multiplication, with {1} as identity element. Conversely, let O be a group. For each 0 E O, we define /} := {((, r/) E O x O 1(0 = 77}, and we set 0 : = { 5 10EO}.Then
7-(o) := (o, 0) is a thin scheme, a Now it is readily verified that, if (X, G) is thin,
T(G(X, G)) ~- (X, O). ~ Moreover, for each group O,
~ T h e elementary proofs of these two facts will be given (iv) in the a p p e n d i x of these notes. W h a t is i m p o r t a n t these two facts allow us to identify each group with its scheme. Therefore, we m a y view the class of groups as a of schemes, namely as the class of thin schemes.
as T h e o r e m A(iii), here for us is that corresponding thin distinguished class
3 This is e..._~y to see. First of all, it is clear that i = l s and that, for each 0 E O, t~* = 0 -1. But also the regularity condition is easily verified for the pair (O, 6). Let fl, 7, e, (, 17 E 19 be given, and assume that (fl,7) E 0- Then, fl~ n 7(" 76 0 if and only if fie = 7( -1 if and only if fie( = 7 if and only if r = r/. Thus, [fl~n 7('1 = ~,r where ~ is the Kronecker delta. 4 In Section 1.7, we shall say what it means for two schemes to be isomorphic.
VII There is still another class of important mathematical objects which can be viewed as a distinguished class of schemes, namely the buildings. Buildings were introduced by J. TITS in [27; (3.1)] as a particular class of chamber complexes. ~ Later, in [28; Theorem 2], Tits characterized the buildings as a particular class of "chamber systems". This characterization indicated already a strong relationship between buildings and schemes. In fact, due to this characterization, it is only a small step to see that, like groups, buildings can be viewed naturally as a distinguished class of schemes. Moreover, the embedding of the buildings into the class of schemes is similar to the one of groups. In other words, there exists a class of schemes which for the buildings plays exactly that role which is played by the class of thin schemes for the groups. Let us give here a rough idea of the definition of this class of schemes. In Section 1.4, we shall define what it means for an element of G to be a "(generalized) involution". The set of all involutions of G will be denoted by Inv(G). For each L C_ Inv(G), we shall define in Section 5.1 what it means for (X, G) to be a "Coxeter scheme with respect to L". Now the pair (X, G) will be called a Coxetev scheme if there exists L C_ Inv(G) such that (X, G) is a Coxeter scheme with respect to L. The Coxeter schemes form the class of those schemes which represent the buildings within the class of all schemes. More precisely, the following three statements hold. First, if (X, G) is a Coxeter scheme, there exists a natural way to construct a building
B(X,G) from (X, G). Secondly and conversely, to each building $, say, there is associated naturally a Coxeter scheme which we shall denote by
Finally, if (X, G) is a Coxeter scheme,
A(z(x, G)) = (x, a). Moreover, for each building 0, =
g.
5 In the hterature, the definition of buildings changes occasionally. In these notes, buildings are always understood to be regular. Viewing buildings as chamber systems (and we shall do that always in this text) we say that a building in the sense of [28] is regularif any two members of one of the defining partitions have the same cardinality. It is an easy exercise in the theory of buildings to prove that buildings in the sense of [27] are regular. It is also obvious that thin buildings in the sense of [28] are regular. In particular, the class of buildings in the sense of the present text contains strictly the class of buildings in the sense of [27].
VllI Occasionally we shall speak of the (B,.A)-correspondence in order to denote the above-mentioned correspondence between buildings and Coxeter schemes. Similarly, the above-mentioned correspondence between groups and thin schemes will be called the (~, T)-eorrespondence. The (B,.A)-correspondence is not as easy to describe as the (9, T)correspondence. Therefore we shall not give the details here. They will be given as Theorem E in the appendix of these notes. There is a simple and natural way to identify Coxeter groups and thin buildings; see [28; (2.3.1)]. Modulo this identification, the (B,A)-correspondence and the (9, T)-correspondence coincide on the class of the thin Coxeter schemes. More precisely, the following two statements are true. If (X, G) is a thin Coxeter scheme, G(X, G) = B(X, G). For each Coxeter group O, T(O) = A(O). As a consequence, thin Coxeter schemes correspond to Coxeter groups. 6 Viewing groups and buildings as the cornerstones of the class of schemes it seems to be promising to develop a structure theory of schemes between these cornerstones. The object of the present lecture notes is in the first place to develop a treatment of schemes analogous to that which has been so successful in the theory of finite groups. As indicated in the first footnote, the condition IX[ E N plays an important role in these notes. Let us say that (X, G) is finite if IX[ E N. The starting point of our approach to schemes is the definition of the complex product. The complex product allows us to treat schemes as algebraic objects. The first chapter is devoted to elementary consequences of the definition of the complex product. Substructures of schemes as well as quotient structures of finite schemes are defined naturally, and we end this introductory chapter with the generalization of the isomorphism theorems [22; w for finite groups due to E. NOETHER. The second chapter begins with a generalization of the fundamental grouptheoretical theorem [16; w due to C. JORDAN and O. H6LDER. As in the theory of finite groups, this theorem allows us to speak of "composition factors". In Sections 2.2, 2.3, and 2.4, we focus our attention on schemes which have thin composition factors. After that, other decompositions of schemes, such as "direct", "quasi-direct", and "semidirect products", are introduced. We include the theorem [9; Theorem 3.17] of P. A. FERGUSON and A. TUR U L L o n "indecomposable" schemes. In the third chapter, we collect various algebraic results which are needed for the representation theory (Chapter 4) and for the theory of generators of schemes (Chapter 5). The fourth chapter gives a general introduction into the representation theory of finite schemes. We start with a generalization of the fundamental group-theoretical theorem [21] of H. MASCHKV.on the semisimplicity of group algebras. After that, our approach is similar to that one given by D. G. 6 ...as is to be expected...
IX HIGMAN in [14]. In this chapter, we restrict ourselves to general structural results ignoring the huge amount of literature which exists in this area. For each L C Inv(G), we shall define in Section 5.1 what it means for (X, G) to be "L-constrained". The pair (X, G) will be called constrained if there exists L C Inv(G) such that (X, G) is L-constrained. In the fifth chapter, we first investigate constrained schemes. The constrained schemes form a class of schemes which is slightly larger than the above-mentioned class of Coxeter schemes. Their definition as well as their treatment seems to be particularly natural. The constrained schemes provide an appropriate framework for showing how smoothly Coxeter schemes are embedded into the class of schemes. From a general algebraic point of view, we consider Section 5.1 as the heart of these notes. Let us call the pair (X,G) thick if {1} = {g E G I ng =- 1}. Then it is particularly easy to exhibit the significance of Theorem 5.1.8(ii). This theorem says in particular that thick constrained schemes and thick Coxeter schemes are the same thing. Therefore, if one is willing to view buildings generally as thick, in other words, if one assumes the definition of [27; (3.1)] for buildings, Theorem 5.1.8(ii) implies that, via the (G, 7-)-correspondence, buildings and thick constrained schemes are the same thing. Since the definition of constrained schemes is particularly simple and natural, Theorem 5.1.8(ii) provides us with probably the most succinct definition of buildings. In the fifth chapter, we include a complete proof of the famous Theorem [8; Theorem 1] of W. FEIT and G. HIGMAN on finite generalized polygons. (We shall present the proof which was given by R. KILMOYER and L. SOLOMON in [20].) Moreover, we give a conceptually alternate and simultaneous approach to the theorems [24; Theorem 2] of S. PAYNE and [23; Satz 1] of U. OTT on polarities of finite generalized quadrangles and finite generalized hexagons via semidirect products. The main focus of the remainder of the fifth chapter is then on an appropriate generalization of Coxeter schemes of "rank" 2. The chapter ends with a generalization of the (algebraic) characterization [34; Theorem B] of finite generalized polygons and Moore geometries; see Theorem 5.8.4. The appendix of these notes is devoted to the embedding of the class of groups as well as the class of buildings into the class of schemes. In other words, we establish explicitly the (G, T)-correspondence and the (/3, .A)correspondence. Let us mention here that, apart from the class of groups and the class of buildings, other classes of mathematical objects as well can be viewed as specific classes of schemes. For the class of distance-regular graphs the embedding was given implicitly by P. DELSARTE in [6; Theorem 5.6]. As a consequence of this, one obtains the (elementary) fact that the class of strongly regular graphs forms a distinguished class of schemes. The fact that Moore geometries can be viewed as a specific class of schemes was partially shown in [34; (2.4)]. In particular, the class of the 2-designs with ~ = 1 forms a
X distinguished class of schemes. As a consequence, a lot of problems and results in graph theory or design theory can be viewed naturally or formulated easily as problems or results on schemes; see, e.g., [32], [33], or [35]. It seems to be a thought-provoking question to ask for other algebraic or geometric objects which can be viewed as classes of schemes. Apart from technical advantages there is at least one other reason for us to view mathematical objects such as groups, buildings, distance-regular graphs, or Moore geometries as schemes. Namely, the language of schemes provides a natural conceptual framework in which the above-mentioned objects may be characterized naturally. Theorem 5.8.4 and Corollary 5.8.5 are characterizations of that type for finite generalized polygons and Moore geometries. A similar characterization of the class of all finite buildings, i.e., a characterization without restriction of the rank, would be a challenging goal. Let us conclude this introduction with three general remarks. First of all, it might be helpful to mention here that the present text is thought to be an introductory monograph. No attempt has been made to give a complete account of the results available on schemes. In particular, the well-worked-out connection between schemes and graphs has been omitted completely. (This connection is discussed extensively in [3] and in [30].) On the other hand, apart from a few elementary facts about vector spaces and groups, the present notes are self-contained. They can be considered as an introduction to the structure theory of schemes. Secondly, the meaning of the symbols Z, IN, and P is fixed for the whole text. We shall denote by Z the set of rational integers. We set
N:= { z ~ z I 0_~H. (iii)
FxH E C(H~H) if and only if F E C.
Proof. (i) Let e E E, let f E F, and let h E H be given. Then, by Theorem 1.5.1(iii), hxn E exnfxH if and only if h E el. This proves the claim. (ii) Let n E 1~ be given. Then, by Theorem 1.5.1(i) and (i),
((F~H)* U F~/4)"
=
((F*)~H U FxH)"
=
((F" U F)~H)" = ((F* U F)")~:H.
Therefore, by Theorem 1.4.1(i),
: U
U ((F* nElv, (U hEN
(F" u
u Fx.)" =
UF)")~H =
=
20
1. Basic Results (iii) By Theorem 1.5.1(i) and (i),
(F~:H)*F,:H = ( r * ):~. F =. = (F*F):~H. Therefore,
F~H
9C(HxH) r162
(F.,,)*F,:. c_ p~. (F*F)~H C F~H r F*F C F r F 9 This proves (iii).
[]
For each g 9 G and, for each H 9 C, we set
gg := {(yH, zH) I z 9 yHgH}.
P r o p o s i t i o n 1.5.3 Let e, f 9 G, and let H 9 C be given. Then the following conditions are equivalent. (a) eII El IH # O. (b) H e l l = n f H . (c) en = fn.
Proof. (a) =:~ (b) Let y, z 9 X be such that (yH, zH) 9 e H rh IH. Then, by definition, z 9 yHeH fq y H f H . Let g E G be such that (y, z) 9 g. Then g E H e l l M H f H . But, by Corollary 1.3.2, G / / H is a partition of G. Thus, Hell = H fH. The implications (b) ::~ (c) and (c) :=~ (a) are obvious.
[]
Let H E C, and let F C_ G be given. Referring to Proposition 1.5.3 we shall write
FIIH := { f . I f ~ F}, although, formally, we defined F // H := { H f H I f 9 F} earlier. Let H, K E C be such that K C_ H and H / / K = G / / K . It follows readily from Proposition 1.5.3 that then H --- G. Occasionally we shall use this elementary observation without reference. For each H 9 C, we set
(X, G) H := ( X / H , G / / H ) .
1.5 Subschemes and Factor Schemes
21
For the remainder of this section, we assume that IX] E N.
T h e o r e m 1.5.4 Let H E C be given. Then we have (i) 1X/H = 1H. (ii) For each g E G, (gH). = (g.)H. (iii) (X, G) g is a scheme. (iv) Let d, e, f E G be given. Then 1
adHeHfH ~ - -nH
E Z abc$. bC=HdHcEHeH
(v) For each g E G, ngHn H ~ nHg H.
Proof. (i) follows from Theorem 1.3.1. (ii) follows from the definition of gH and from Lemma 1.2.5(i). (iii), (iv) By Proposition 1.5.3, G / / H is a partition of X / H • X / H . From (i) we know already that 1X[H E G//H. Moreover, from (ii) we know that, for each g E G, (gH). E G//H. Thus, in order to prove (iii), is suffices to verify the regularity condition for (X, G) H. We compute the structure constants explicitly, so that also (iv) will be proved. Let d, e, f E G, and let y, z E X be given. Assume that (yH, zH) E fH. Then, by Theorem 1.3.1, we may even assume that (y, z) E f. Define W := yHdH f3 zHe*H. Then we have
Iwl=
abcl bEHdH cEHeH
Also
x CW r
xH E (yH)(d H) M (zH)((e")*).
Thus, the number
I(yg)(d H) f3 (zH)((eH)*)I -- IWI nH does not depend on the choice of the pair (yH, zH) E fH. (v) follows from (iv) and from Lemma 1.2.5(iii).
[]
We do not know whether or not the hypothesis IX] E N can be deleted in Theorem 1.5.4. For each H E C, (X, G) H will be called the factor scheme of (X, G) over H.
22
1. Basic Results
Let p E II~, and let F C G be given. We shall say that F is p-valenced if, for each f E F, n/ is a power o f p .
L e m m a 1.5.5 Let p E I?, and let H E C be given. Assume that IX[ i s a power of p and that H is thin. Then G is p-valenced if and only if G / / H is p-valenced.
Proof. Let g E G be given. Then, by Theorem 1.5.4(v), n g H n H ~ TIHg H .
For each f E HgH, we conclude from Corollary 1.2.9(ii) that n/ = ng. (Recall that H is assumed to be thin.) Therefore, nHgH : ]HgHing, so that we have n g H n H --~
]HgHing.
By hypothesis, IX] is a power of p. Moreover, by Theorem 1.3.6(iv), IX[ =
nHIX/H]. Therefore, nH is a power of p. We wish to prove that [HgH] is a power of p, too. By hypothesis, H is thin. Therefore, IHI = nil, so that H is a power o f p . From Corollary 1.2.9(ii) we deduce that, for each x E X, the p-group
g((x,
• G((x,
acts transitively on HgH. 2 Thus, IHgHI is a power of p. Now the last equation says that ng is a power of p if and only nan is a power of p. Since g E G has been chosen arbitrarily, we have shown that G is pvalenced if and only if G / / H is p-valenced. []
1.6 Computing in Factor Schemes Throughout this section, we assume that IX] E N.
T h e o r e m 1.6.1 Let n E N \ {0}, and let F1, ..., Fn C_ G be given. Then, for each g E a and, for each H E C, gH E ( F 1 / / H ) ' " (Fn//H) if and only if g E ( H F 1 H ) . . . (HF, H). 2 Recall that 9((X, G)xH) denotes the group which is associated to the thin scheme (X, G)~H via the (~, T)-correspondence.
1.6 Computing in Factor Schemes
23
Proof. If n = 1, the claim follows immediately from Proposition 1.5.3. Therefore, we may assume that 2 < n. Assume first that gH E (F1//H)...(Fn//H). Then there exist e, f E G such that eH E (F1//H)... (F,_I//H), f H E Fn//H, and a~nyng, r O. Since aeHlHgH ~ O, there exist b E Hell and c E H f H such that abca r 0; see Theorem 1.5.4(iv). Thus, by definition,
g E (HeH)(HfH). On the other hand, by induction, eu E (F1//H)...(Fn-~//H) yields e E (HF1H)... (HFn_IH). Moreover, f g E F , / / H implies that f E HF, H; see Proposition 1.5.3. Therefore,
(HeH)(HfH) C (HF1H)... (HF,~H). Together, we obtain that g E (HF1H)... (HF,,H). Conversely, assume that g E (HF~H)... (HF,~H). Then there exist e E (HF1H)...(HF,~_IH) and f E HFnH such that g E ef. Since g E e f, a~fg r O. Thus, by Theorem 1.5.4(iv), a~,lHgn r O. This means that
gH E eH f H. On the other hand, by induction, e G (HF1H)...(HFn-IH) yields en E ( F 1 / / H ) ' " (Fn-1//H). Moreover, f E HF,~H implies that fHE F.//H; see Proposition 1.5.3. Thus, by Lemma 1.2.1(i),
eH f H C_ ( F 1 / / H ) . " (F,,//H). It follows that g n E ( F i f t H ) . . . (F~//H).
[]
Let H E C, and let F C_ G be such that H F H C F. Then F//U E C(G//H) if and only if F E C. C o r o l l a r y 1.6.2
Proof. Assume first that F//H E C(G//H). Let g E F ' F be given. Then g E (HF*H)(HFH). Thus, by Theorem 1.6.1 and Theorem 1.5.4(ii), gH E ( F ' / / H ) ( F / / H ) : (F//H)*(F//H) C_ F//H. Now Theorem 1.6.1 yields g E H F H C_ F. Since g E F*F has been chosen arbitrarily, we have shown that F E C. Conversely, let us now assume that F E C. Let g E G be such that gH E (F//H)*(F//H). By Theorem 1.5.4(ii), we have (F//H)* = F*//H. Therefore, gH E (F*//H)(F//H). Thus, by Theorem 1.6.1,
g E (HF*H)(HFH) = (H*F*H*)(HFH) C_ F*F C F. It follows that gH E F//H.
24
1. Basic Results
Since gH E (F//H)* (F//H) has been chosen arbitrarily, we have shown [] that F N H E C(G//H).
C o r o l l a r y 1.6.3 Let g E G, and let H E C be given. Then UgH = 1 if and only if g*Hg C H.
Proof. First of all, by Theorem 1.5.4(iii) and Lemma 1.2.6, nan
--
1 r
{l."}
--
(gH).gH.
Assume now that rig, -= 1, and let f E g*Hg be given. Then, by Lemma 1.2.1, f E (Hg*H)(HgH). Thus, by Theorem 1.6.1 and Theorem 1.5.4(ii),
Y" E ( g ' ) ' ( g ' ) = ( g " ) ' g " = {1"}. It follows that fH = 1H. Thus, by Proposition 1.5.3, f E H. Since f E g*Hg has been chosen arbitrarily, we have shown that g*Hg C H. Conversely, assume that g*Hg C_ H, and let f E G be such that fH E (gtt).gH We shall be done if we succeed in showing that fH = 1H. From Theorem 1.5.4(ii) we know that (gH). = (g,)H. Therefore, we have fH E (g,)HgH. Thus, by Theorem 1.6.1 and Lemma 1.2.1(ii), f E (Hg*H)(HgH) C_H. Therefore, by Proposition 1.5.3, fH = 1H. []
T h e o r e m 1.6.4 Let H E C, and let F C_ G be such that H C_ (F). Then (FIIH) = (F)IIH.
Proof. Let g E G be such that gH E (F//H). Then, by Theorem 1.4.1(ii), there exists n E 1~ such that gH E (F//H) n. Thus, by Theorem 1.6.1, g E (HFH)". Now, as we assume that H C_ (F), g E (F); see Theorem 1.4.1(ii). It follows that gH E (F)//H. Conversely, let g E G be such that gH E (F)//H. Then there exists f E (F) such that gH = fH. Since f E (F), there exists n E 1~ such that f E Fn; see Theorem 1.4.1(ii). Since gH = fH, g E H f H ; see Proposition 1.5.3. It follows that g E HFnH C (HFH) ~. Therefore, by Theorem 1.6.1, gH E (F//H) '~ C_ (F//H); see Theorem 1.4.1(ii) once again. D T h e o r e m 1.6.5 Let H E C, and let F C_G be such that (F//H) = G//H.
Then (F U H) = G.
1.6 Computing in Factor Schemes
25
Proof. From ( F f f H ) = G f l H we obtain that ( F ) / / H = GffH; see Theorem 1.6.4. Let g E G be given. Then there exists f E (F) such that gH = fH. [] Therefore, by Proposition 1.5.3, g E H f H C_ U ( F) U C_ ( F U H).
C o r o l l a r y 1.6.6 Set H := P(G). Let g E G be such that (gH) = G / / H . Then (g) = G.
Proof. This follows from Theorem 1.6.5 and Theorem 1.4.4(i).
[]
The following theorem describes a relationship between factor schemes of subschemes of (X, G) and subschemes of factor schemes of (X, G).
T h e o r e m 1.6.7 Let x E X be given. Let H, K E C be such that K C H. Then we have (i) K,H E C(H~:H), and ((X, G)~H) K*" is a scheme. (ii) H / / K E 6 ( G / / K ) , and ((X, G)K)(~K)(H//g) is a scheme. (iii) ( ( X , U)xH) "' ~K,, : ((X, G)K)(xK)(H//K).
Proof. (i) Since K E C, K~:H E C(HxH); see Lemma 1.5.2(iii). Therefore, by Theorem 1.5.1(ii) and Theorem 1.5.4(iii),
is a scheme. (ii) Since H E C, S I l K E C(G//K); see Corollary 1.6.2. Therefore, we deduce from Theorem 1.5.4(iii) and Theorem 1.5.1(ii) that
((x, is a scheme. (iii) First of all, we have
xH/I'G:H = {yI'( l Y E xH} = ( x K ) ( H / / K ) . But, for each h E H we also have,
(hxH) g~n --_ (hK)(xK)(H//K), whence
HxH//KxH = I h E H) =
26
1. Basic Results
{(h K )(~m(-//~) I h ~ H}
=
(H//K)(~:K)(H//K). It follows that
((x, a)~.) K=. = ((x, a)K)(.m(.//u). []
Let x E X be given. Let H, K E C be such that K _C H. For the remainder of these notes, we abbreviate
(x, c)~.I< := ( ( x , c ) ~ . ) ~='.
1.7 Morphisms Let (W, F) be a scheme, and let r
XUG
-4 W U F
be a map with X r C W and Gr _C F. Then r is called a morphism from (X, G) to (W, F) if, for all y, z E X and, for each g E G,
(y, z) E g ~
(yr zr E gr
Let us first collect several elementary facts about morphisms.
L e m m a 1.7.1 Let r be a morphism from (X, G). Then (i) For each g E G, g*r = (gr (ii) For all e, f E G, (ef)r C_ eCfr (iii) For each F C_ G, (F)r C_ (Fr
Proof. (i) Let g E G be given. Let y, z E X be such that (y,z) E g. On the one hand, (y, z) E g implies that (z, y) E g*, whence (zr yr E g*r On the other hand, (y, z) E g implies also that (yr162 E gr Therefore, (zr162 E (gr It follows that g*r f-)(gr ~ ~, so that g*r = (gr (ii) Let e, f E G, and let g E e f be given. Then, by definition, a~sg ~ O. Thus, there exist y, z E X such that (y,z) E g and ye n z f * ~ 0. Let x Eye (1 z f* be given.
1.7 Morphisms
27
Since (y, x) 9 e, (y4', x4') 9 e4'. Similarly, as (x, z) 9 f, (x4', z4') 9 re. But (y4,, z4,) 9 g4,, since (y, z) 9 g. Thus, a~r r ~- O. It follows that gr 9 e4,f4'. Since g 9 e f has been chosen arbitrarily, we have proved that (el)4' C_ e4'f4'. (iii) follows from (i) (ii), and from Theorem 1.4.1(i). [] L e m m a 1.7.2 Let (W, F) be a scheme, and let 4' be a morphism from (X, G) to (W, F). Then we have (i) 14' = 1w. (ii) For each E 9 C(F), E4' -1 9 C. (iii) Assume that 4' is surjective. Then, for all d, e 9 F, (de)4' -~ = d4'- l e r -1. Proof. (i) Let z E X be given. Then (x, x) E 1. Therefore, (xr x4,) 9 1r On the other hand, as x4, 9 W, (z4,, xr 9 l w . It follows that 14, = l w . (ii) Let E 9 C(F) be given. Then 1w 9 E. Thus, by (i), 14' 9 E, which means that 1 9 E r In particular, E4,-1 ~= 0. Let c, d 9 E4, -1, and let g 9 c*d be given. Then
g4, 9 (c'd)4' C__c'4'd4' = (c4')*d4' C_ E*E C_ E; use Lemma 1.7.1(i), (ii). It follows that g 9 E4' -1. (iii) Let d, e 9 F, and let g 9 (de)4' -1 be given. Let y, z 9 X be such that (y, z) 9 9. Then (y4', z4') 9 94' 9 de. Thus, by Lemma 1.2.4, there exists w 9 y4'd such that z4' 9 we. Since 4' is assumed to be surjective, there exists x 9 X such that x4' = w. Let b 9 G be such that (y,z) 9 b, and let c 9 G be such that (x, z) 9 c. Then (y4', xr 9 54, and (z4,, z4,) 9 c4,. It follows that 54' = d and c4' = e. Thus, g 9 bc C d4'-ler -1. Conversely, let g 9 dr -1 be given. Then there exist b 9 d4' -1 and c 9 er -1 such that 9 9 bc. Thus, by Lemma 1.7.1(ii), g4, 9 (bc)4, C b4'e4' = de, whence g 9 (de)4' -1.
[]
Let (W, F) be a scheme, and let 8 be a morphism from (X, G) to (W, F). Then we set r := r x W) and r
F).
Note that r (respectively, Ca) is not just the restriction of 4, to X (respectively, G). Also the codomain has changed.
28
1. Basic Results
L e m m a 1.7.3 Let r be a morphism from (X, G). Then (i) If Cx is surjective, Ca is surjective. (ii) / f Ca is injective, Cx is injective. (iii) Assume that Ca is injeetive. Then, for all y, z E X and, for each g E G, (yr zr E gr implies that (y, z) E 9.
Proof. (i) Let (W, F) be the scheme such that r is a morphism to (W, F). Let d E F be given. Then there exist t, u E W such that (t, u) E d. Since Cx is assumed to be surjective, there exist y, z E X such that yr = t and zr = u. Let 9 E G be such that (y, z) E g. Then, as r is assumed to be a morphism, (yr zr E 9r Therefore, (t, u) E 9r It follows that gr = d. (ii) Let y, z E X be such that yr = zr Then, by Lemma 1.7.2(i), (yr162 E 1r Let g E G be such that (y,z) E g. Then, as r is assumed to be a morphism, (yr zr E 9r It follows that 1r = 9r Thus, as Ca is assumed to be injective, we must have 1 = 9. In particular, y = z. (iii) Let y, z E X, and let 9 E G such that (yr zr E 9r be given. Let f E G be such that (y, z) E f. Then (yr zr E fr Therefore, f r = gr But, by hypothesis, Ca is injective. Therefore, f = 9. It follows that (y, z) E g. []
A morphism r from (X, G) is called a fusion if Cx is bijective. We shall not investigate fusions here. They will be used later to define quasi-direct products; see Section 2.6. A morphism r from (X, G) is called a homomorphism if, for all y, z E X and, for each g E G, (yr162 E 9 r
~
3v, w E X : v r 1 6 2
wr162
(v,w) Eg.
Note that, by Lemma 1.7.3(iii), an injective morphism from (X,G) is automatically a homomorphism. Let r be a homomorphism from (X, G). Then we set kerr := {9 E G 19r = 1r The set ker r will be called the kernel of r
T h e o r e m 1.7.4 Let r be a homomorphism from (X, G). Then the following conditions are equivalent. (a) Cx is injective. (b) r is injective. (c) Let y, z E X, and let g E G be such that (yr zr E gr Then (y, z) E g. (d) kerr = {1}.
1.7 Morphisms
29
Proof. (a) ==~ (b) Let e, f E G be such that er = f r Let y, z E X be such that (y, z) E e. Then, as er = r e , (yr zr E fr Thus, as r is assumed to be a homomorphism, there exists v, w E X with vr = yr we = zr and (v,w) E f. Since vr = yr v = y; see (a). Similarly, as wr = zr w = z. Therefore, (y, z) E f. It follows that e = f. (b) =~ (c) follows immediately from Lemma 1.7.3(iii). (c) =~ (d) Let g E ker r be given. Then, by definition, gr = 1r Let y, z E X be such that (y,z) E g. Then, as r is a morphism, (yr162 E gr It follows that (yr zr e 1r Thus, by (c), (y, z) E 1, whence g = 1. Since g E ker r has been chosen arbitrarily, we have proved (d). (d) :=~ (a) Let y, z E X be such that yr = zr Then, by Lemma 1.7.2(i), (yr zr E 1r Let g E G be such that (y, z) E g. Then, as r is a morphism, (yr162 E gr It follows that gr = 1r Thus, by definition, g E kerr Now (d) forces g = 1. In particular, y = z. []
Let (W, F) be a scheme. (X, G) and (W, F) will be called isomorphic if there exists a bijective homomorphism from (X, G) to (W, F). In this case, we shall write (X, G) ~ (W, r ) . For the next two theorems, we assume that IXI E H. T h e o r e m 1.7.5 [HOMOMORPHISMTHEOREM](i) Let H E C be given. For each x E X, we set xr := xH; for each g E G, we set gr := gH. Then r is a surjective homomorphism from (X, G) to (X, G) H which satisfies H = ker r (ii) Let r be a homomorphism from (X, G). Then ker r E C. Moreover, (Xr (Gr162 is a scheme with (Xr (Gr162 ~ (X, G) kerr
Proof. (i) Let y, z E X, and let 9 E G be such that (y, z) G g. Then, as 1 E H, z E yg C yHgH. Thus, by definition, (yH, zH) E gH, which means that (yr zr E gr Since y, z E X and g C G have been chosen arbitrarily, we have shown that r is a morphism. Now let y, z E X, and let g E G be such that (yr zr C gr Then, by definition, (yH, zH) E gH which means that z E yHgH. Thus, by Lemma 1.2.4, there exist v, w E X such that v E yH, w E zH, and (v,w) E g. By Theorem 1.3.1, v E yH is equivalent to vH = yH, and this means that vr = yr Similarly, we deduce that we = zr Thus, r is even a homomorphism from (X, G) to (X, G) u. It follows immediately from the definition of r that r is surjective. By definition, g E kerr if and only if g g = 1F. By Proposition 1.5.3, gH = 1F if and only if g C H. Thus, H = ker r
30
1. Basic Results
(ii) Set H := ker r Then, by Lemma 1.7.2(i), (ii), H E C. Let y, z E X, and let g E G be such that (y,z) E g. Then (yr162 E gr Thus, by Lemma 1.7.20), yr162
r
gr162
gEH
r
r
yH=zH.
In particular,
tx : X I H
-+ Xr
xH ~+ xr
is a bijection. From this, we obtain that, for all e, f E G, er : f r
3v, w,u, z E X : v r 1 6 2 3v, w,y, z E X : v H = y H ,
wr162
(v,~)ee, (~,z) ES r
wH=zH,
Hell = HfH
r
eH
(v,w) Ee, (y,z) E f =
r
fH;
see Lemma 1.2.4 for the third and Proposition 1.5.3 for the fourth equivalence. Thus, tG : G / / H -+ (Gr162 gH ~ (gr162 is a bijection. Let us now prove that tx U tc is a morphism. (From Lemma 1.7.3(iii) we know that then r is even a homomorphism.) Let y, z E X, and let g E G be such that (yH, zH) E gH. Then, by definition, z E yHgH. Thus, by Lemma 1.2.4, there exist v, w E X such that v E yH, w E zH, and (v,w) E g. Since r is a morphism, (v, w) E g implies that (re, we) E gr On the other hand, as v E yH, vH = yH, whence vr = yr Similarly, w E zH implies that we = zr Thus, we conclude that (yr zr E gr []
The homomorphism r in Theorem 1.7.5(i) will be called the natural ho-
momorphism from (X, G) to (X, G) H. T h e o r e m 1.7.6 [FIRST ISOMORPHISMTHEOREM]Let H, K E C be such that K C_ H. Then ((X, G)K) HIIK ~- (X, G) H.
Proof. Let us denote by r (respectively, ~b) the natural homomorphism from (X, G) to (X, G) K (respectively, from (X, G) K to ((X, G)K)H//K). Then r 1 6 2is a surjective homomorphism. On the other hand, we know from Theorem 1.7.5(i) that H / / K = ker r Therefore, we have ker(r162 = {g ~ G l a0 E H / / K } = {g E C I g ~ E H / / K } = H;
1.7 Morphisms
31
use Proposition 1.5.3 for the third equation. Now the claim follows from Theorem 1.7.5(ii). []
Let E, F C G be given, and assume that F • 0. We set NE(F) := {e E E ] Fe = eF}. NE(F) will be called the normalizer of F in E. Let H E C be given. Clearly, from the definition of N c ( H ) we deduce immediately that H C_ NG(H). But, in general, we do not have that N a ( H ) E C. In particular, the first claim of the following Lemma needs a proof.
L e m m a 1.7.7 Let H E C be given. Then (i) N a ( H ) H C_ No(H). (ii) Let F C_ G be such that U A F r 0. Then NH(F) C_ N H ( H N F).
Proof. (i) Let g E N a ( H ) H be given. Then there exists e E NG(H) such that g E ell. Since e E No(H), He = ell. Therefore, g E He. Thus, by Theorem 1.3.1, Hg = He = e H = gH. It follows that g E No(H). (ii) Let h E NH(F) be given. Then, by Lemma 1.3.5(i), h ( H A F) = H M h F = H M F h = (H A F ) h . []
Let H, K E C be such that K C_ H. K is said to be normal in H if H _C NG(K). In this case, we shall write K hacks