The Unitary and Rotation Groups

THE UNITARY AND ROTATION GROUPS by FRANCIS D. MURNAGHAN Consultant, Applied Mathematics Laboratory The David Taylor Mod...

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THE UNITARY AND ROTATION GROUPS

by FRANCIS D. MURNAGHAN Consultant, Applied Mathematics Laboratory The David Taylor Model Basin

SPARTAN BOOKS

WASHINGTON, D. C.

1962

Library of Congress Catalog Card No. 62-19096 Copyright ®1902 by Fns xczs D. MURNA(n1AN

Printed in the United States of America All rights reserved This book or parts thereof, may not be reproduced in any form without permission of the publishers

LECTURES ON APPLIED MATHEMATICS VOLUME III

Volume I

The Laplace Transformation

Volume II The Calculus of Variations

Volume III The Unitary and Rotation Groups

Preface

This book, third in a series, is based on lectures on applied mathematics given at the Applied Mathematics Laboratory of the David Taylor Model Basin. It treats the n-dimensional unitary and rotation groups and their finite-dimensional representations. The element of volume of these groups is derived and the concept of integration over their parametric spaces is treated in detail. A modification of Euler's parametrization of the 3-dimensional rotation group is given, and this modification is extended to the n-dimensional rotation group where n is any integer greater than 3. A similar parametrization of the n-dimensional unitary group is furnished and the close connection between the 2-dimensional unimodular unitary group and the 3-dimensional rotation group is clearly explained. This connection gives rise to the spin representations of the 3-dimensional rotation group which are, precisely speaking, representations of the 2-dimensional unimodular unitary group rather than of the 3-dimensional rotation group. These spin representations of the 3-dimensional rotation group, and also the spin representations of the proper, time-sense preserving Lorentz group, are treated in considerable detail.

As in the preceding two volumes, care has been taken to make the lectures self-contained by giving detailed proofs of the basic theorems. It is hoped that the material will prove useful to workers in theoretical physics in view of the increasing importance of rotation groups of higher dimension than 3. Washington, D. C. May 2/, 1962

FRANCIS D. MURNAGIIAN

Contents Preface ...................................................

1. The n-Dimensional Unitary and Rotation Groups ..............

vii 1

2. The Parametrization of the n-Dimensional Unitary and Rotation Groups...................................................

7

3. The Class and In-Class Parameters of the n-Dimensional Unitary Group ....................................................

14

4. The Class and In-Class Parameters of the n-Dimensional Rota-

tion Group ................................................

5. The Element of Volume of a Parameter Group ................. 6. The Characteristic Matrices of a Matrix Group ................ 7. The Adjoint Representation of the n-Dimensional Unitary Group. 8. The Class Factor of the Element of Volume of the n-Dimensional

Unitary and Rotation Groups ...............................

20 27 33 40 47

9. Representations of the n-Dimensional Unitary and Rotation

Groups ...................................................

55

10. The Orthogonality Relations for a Compact Parameter Group... 63 11. The Irreducible Representations of the n-Dimensional Unitary

Group ....................................................

71

12. The Analysis of any Power of a Representation of the n-Dimensional Unitary Group 80 13. The Spin Representations of the Proper, Time-Sense Preserving Lorentz Group and of the 3-Dimensional Rotation Group ...... 88 14. The Simple Characters of the n-Dimensional Rotation Group.... 98 15. The Analysis of the Representations Tcw,> ® (a) of the 2-Di-

.......................................

mensional Unimodular Unitary Group ........................ 107 16. The Analysis of the Product of Irreducible Representations of

............................ 119

the n-Dimensional Unitary Group 17. The Modification Rules for the n-Dimensional Rotation Group.. 129

18. The Analysis of the Representations r (p) ® (a) and r«, 0 (X) of the n-Dimensional Unitary Group

.........................

138

Bibliography .................................................. 147 Index ........................................................ 149

1

The n-Dimensional Unitary and Rotation Groups

Let x, ,

-

-

, x" be any set of n linearly independent n X 1 matrices whose

elements are complex numbers, n being any positive integer. Then the n-dimensional matrix X whose kth column-matrix is xk, k = 1, , n, is nonsingular, and we say that X determines a basis for complex vectors in n-dimensional space. When X is the n-dimensional identity matrix E. , we refer to the basis which is determined in this way as the standard basis, and we say that the elements of xk , k = 1, , n, are the coordinates, with respect to the standard basis, of the kth vector of the basis which is determined by X. If c is any n X 1 matrix whose elements are complex numbers, we have Xc = x,cl + + x"c", and we say that the elements of c are the coordinates, with respect to the basis which is determined by X, of the vector whose coordinates, with respect to the standard basis, are furnished by the elements of the n X 1 matrix Xc. Thus an n-dimensional complex vector requires for its specification two things: (1) A nonsingular n-dimensional matrix, X, whose elements are complex numbers; the role of X is to determine a basis for our n-dimensional linear vector space. (2) An n X 1 matrix, c, whose elements are complex numbers; the role

of c is to furnish the coordinates, with respect to the basis which is determined by X, of the vector we wish to specify. If X' is any other nonsingular n-dimensional matrix, whose elements are complex numbers, and c' is an n X 1 matrix, whose elements are complex numbers, then the co-

ordinates, with respect to the standard basis, of the vector whose coordinates, with respect to the basis which is determined by X', are the I

2

Lectures on Applied Mathematics

elements of c', are furnished by the elements of the n X 1 matrix X'c'. Denoting by (X, c) the vector whose coordinates, with respect to the basis which is determined by X, are the elements of c, we have the following result:

The necessary and sufficient condition that the vectors (X, c), (X', Cl) should be the same is the equality of the two n X 1 matrices Xc and X'c' or, equivalently, the existence of the relation c' = Sc, where S is the nonsingular n-dimensional matrix (X')-1X.

Now let (X, cl) and (X, cg) be any two vectors in our n-dimensional linear vector space, and let us consider the 1-dimensional matrix c, c2 , where the star attached to any matrix denotes the transposed conjugate. We term this 1-dimensional matrix, or complex number, the numerical product, with respect to the basis which is determined by X, of the vector (X, cg) by the vector (X, c,) and we observe that this numerical product is not, in general, a symmetric function of the two vectors (X, c,) and (X, cg). Indeed, since the star of the product of any two complex matrices is the product, in reverse order, of the stars of the two factor matrices, and since the star of a 1-dimensional matrix is its complex conjugate, the necessary and sufficient condition for the equality of ct*cl and cl*c2 is the reality of cl*cg . If, in particular, c; c2 is zero, so also is cg*c, . If we were working in the real field, i.e., if the elements of the matrices X, c, and CS were all real, then c2*cl would always be the same as c,*c2 , and so we term the numerical product, with respect to the basis determined by X, of the real vector (X, cg) by the real vector (X, c,) or, equivalently, the numerical

product, with respect to the basis determined by X, of the real vector (X, c,) by the real vector (X, cg), simply the symmetric product of these two vectors. In general, we term the numerical product with respect to the

basis determined by X, of the vector (X, c) by itself, namely c*c, the squared magnitude, with respect to the basis determined by X, of the vector (X, c). It is clear that this squared magnitude is a nonnegative real number, being zero only when c is the zero n X 1 matrix. When c is the zero n X 1 matrix, we say that (X, c) is the zero vector, it being evident from the equality criterion Xc = X'c' that the vectors (X, 0) and (X', 0) are the same, no matter what are the nonsingular n-dimensional matrices X and X'. It is easy to see that the numerical product of the vector (X, cg) by the vector (X, cl) depends, in general, on the basis used to present the two vectors. Indeed, the coordinates, with respect to the standard basis, of the vectors (X, c,) and (X, cg) are furnished by the elements of the two n X t matrices Xe1 and Xcg , respectively, so that the numerical product, with respect to the standard basis, of the vector (X, c2) by the vector (X, cl) is (Xc,) *Xcg = c,*X*Xcg , and the necessary and sufficient condition that

The Unitary and Rotation Groups

S

this be the same as c,*c2 , no matter what the n X 1 matrices c, and c2, is that X*X be the n-dimensional identity matrix E. :

X*X=E We term any n-dimensional matrix, X, which satisfies this equation an n-dimensional unitary matrix, and we reserve the symbol U for any such matrix so that

U*U=E,, On taking the determinants of the matrices on each side of this equation

we see, since dot U* = (det U) * = det U, where the superposed bar denotes the complex conjugate, that det U is a complex number, exp i¢, #

real, of unit modulus. When the n-dimensional matrix X is a unitary matrix U we term the basis determined by U a unitary basis. Whether or not the nonsingular n-dimensional matrix X is unitary, we

consider the correspondence (E , c) - (X, c) = (E , Xc) between the two vectors whose coordinates with respect to the standard basis and the basis determined by X, respectively, are furnished by the elements of the arbitrary n X 1 complex matrix c. Using the standard basis, this correspondence is specified by the relation c -- c' = Xc and induces the transformation c, c2 -+ ci*X*Xcq of the numerical product, with respect to the

standard basis, of the vector (E , c2) by the vector (E , c,). Thus the necessary and sufficient condition that this numerical product should be

insensitive to the transformation in question, no matter what the two vectors (E. , c,) and (E. , ce), is that X be a unitary n-dimensional matrix. It follows that the product U2U1 of any n-dimensional unitary matrix,

U1, by any n-dimensional unitary matrix, U2, is unitary; indeed, the result of following the transformation c - c' = Ulc by the transformation c' --)' c" = Usc is the transformation c -* c" = U2U1c, and ci*c2 is insensitive to this transformation no matter what the two n X 1 complex matrices ci and cs. Similarly, the reciprocal of any unitary n-dimensional matrix is unitary; for the transformation c -> c' = Uc may be written in the form U-'c -c and the insensitiveness of the numerical product, no matter what

the n X I matrices cl and c2 , proves that U' is unitary. The relation U*U = E. shows that U`' = U*. The n-dimensional identity matrix is unitary and so the collection of all n-dimensional unitary matrices constitutes a group which is known as the n-dimensional unitary group. An important subgroup of the n-dimensional unitary group consists of those n-dimensional unitary matrices whose determinants are unity; we term any square matrix whose determinant is unity unimodular, and accordingly this subgroup is known as the n-dimensional unimodular unitary group. The subgroup of the n-dimensional unitary group which consists


of those unitary matrices all of whose elements are real is known as the n-dimensional orthogonal group, an n-dimensional orthogonal matrix being

a real n-dimensional unitary matrix. Since det U = exp ii, 46 real, the determinant of any n-dimensional orthogonal matrix is either 1 or -1. The subgroup of the n-dimensional orthogonal group which consists of all unimodular n-dimensional orthogonal matrices or, equivalently, the subgroup of the n-dimensional unimodular unitary group which consists of all real n-dimensional unimodular unitary matrices, is known as the n-dimensional rotation group, an n-dimensional rotation matrix being a real n-dimensional unimodular unitary matrix. If (u, , , u,) are the column-matrices of an n-dimensional unitary matrix the equation U*U = E. is equivalent to the n equations uj*u, = 1, j = 1, - , n, plus the n(n - 1)/2 equations uj*uk = 0, j < k, it being unnecessary to write the equations uk'u, = 0 since the equation a j*uk = 0 implies, on starring both sides, the equation uk*uj = 0. Since u,'uj is the sum of the squared moduli of the elements of u, , it follows that no element of an n-dimensional unitary matrix can have its modulus > 1. Furthermore, if any element of uj has its modulus = 1, all the other elements of u j are zero. On applying this argument to U*, which is unitary when U is, we see that if any element of U has its modulus = 1, then all the elements of U that lie in the same row or column as this element are zero, the row-matrices of U being the conjugates of the column-matrices of U*.

Exnar'LE 1. n = I The typical 1-dimensional unitary matrix is exp ii, -7r < ¢ < a. Thus the I-dimensional unitary group is a 1-parameter group, the parametric space being furnished by the relations -ir < 0 < ir. This parametric space is not, as it might appear to be, an interval closed at one end and open at the other. We must identify the two end-points, -ir and zr, of this interval, and we express this fact by the statement that the parametric space has the topological character of the circumference of the unit circle. A function of the 1-dimensional unitary group is not merely a function of 0 but rather a periodic function, of period 2w, of 0. The fact that the parametric space is bounded and closed (the closure being effected

by the identification of the two end-points of the interval -u < 0 < r) assures us that any continuous function of the 1-dimensional unitary group, i.e., any function of the group which is a continuous function of the parameter ¢, is bounded over the group. We shall see shortly the fundamental importance of this fact. The 1-dimensional orthogonal group consists of the two elements I and -1 and the 1-dimensional rotation group consists of the single element 1.


6

EXAMPLE 2. n = 2

Writing U =

Ib

d

jwehaveda+&b=j'aC+dd=1'CW -}- 6d = 0.

la where k = det U = exp 15, -7r < 8 < ar. If b ;d 0 and a = 0, d = 0, and U is of the form -k6 I where k = det U = exp ib. If neither a nor b is 0, c = - k6, (0 a d = ka and U is of the form I 6 where k = det U = exp ib. On If b= 0, 1 a I= 1, c = 0 and U is of the form

I

I

a

b

writing U in the form exp (0/2) U, , U, is unimodular and, hence, of the form

I

b

a6l where da + 6b = 1. We may write a = cos # exp ia,

b = sin 0 exp ip where - (ar/2) < a < ,r/2, - (a/2) < S < ,r/2 and - IF < 0 < IF, cos 4, being positive or negative according as arg a lies in the

interval - (7/2) < arg a< 7r/2 or not, and sin 0 being positive or negative according as arg b lies in the interval - (ar/2) < arg b < ar/2 or not. In order to emphasize the difference between the angles a and p and the angle 0 we refer to a and p as latitude angles and to 0 as a longitude angle. Thus the typical 2-dimensional unimodular unitary matrix is of the form cos 4, exp is + sin 0 exp -ip sin 4, exp ip

cos 0 exp -ia

where 0 is a longitude angle and a and O are latitude angles, and the 2-dimensional unimodular unitary group is a 3-parameter group whose parametric space is defined by the relations

-7r < 0 < 7;

-2 < a
y = ax, is the left-translation of the parametric space which is

induced by the left-translation X - Y = AX of the group. We assume that not only is y a continuous function of x and a, but also that the Jacobian matrix y, exists and is a continuous function of x and a; since a = yx ', this implies that yx is a continuous function of x, x-' and y. We assume, further, that z7', the point of the parametric space which corresponds to the element X-' of our parameter group, is a continuous function

of x so that yx is a continuous function of x and y which we denote by J(x, y) Y. = J(x, Y) In this relation x and y may be any two freely chosen nonexceptional points, say x, and y, , of the parametric space. Then a = y,xi ' and y, is


$9

determined at x = x, and y = y, from the relation y = y,x,-'x by differentiation and subsequent evaluation at x = x, and y = y, . In particular, when y, = x, , we have y = x, so that J(x, , x,) is the r-dimensional identity matrix E, and, since x, may be freely chosen, we may write this result in the form

J(x, X) = E, It is easy to we that the independent variables x and y are separable in the function J(x, y), this property being a reflection of the group property possessed by the elements X of our parameter group. Indeed, if we follow

the left-translation X - Y = AX of our parameter group by the lefttranslation Y -- Z = BY of the group, we obtain the left-translation X - Z = CX, where C = BA, of the group. Then the chain rule, z, = zsys

,

of differential calculus tells us that J(x, z) = J(y, z) J(x, y). In this relation the nonexceptional points x, y, z of the parametric space may all be freely chosen, the choice of x and y determining a and the choice of y and z determining b. On making this choice so that z = x we see that, no matter what are the two nonexceptional points x and y of the parametric space, J(x, y) is nonsingular, its reciprocal being J(y, x). Thus zs = {J(z, y)1-' J(x, Y)

Holding y fixed and allowing x and z to vary independently over the no

exceptional points of the parametric space, J(x, y) is a function of the single variable x which we denote simply by J(x) ; then

J(x, z) = (J(z)r-' J(x) proving the separability of the variables x and y in the function J(x, y). We shall term the nonexceptional point y of the parametric space which we have agreed to hold fixed the base-point of the parametric space. At

this point J(x) becomes the r-dimensional identity matrix, E,, since

J(y) = J(y, Y) = E, . The Jacobian determinant, d(z), is such that d(z) = (det zs)d(x) and so, since det zs = jdet J(z)r' det J(x), we have (let J(z) d(z) = det J(x)d(x) Since J(x) is continuous and nonsingular over the connected parametric space, both dot J(x) and dot J(z) have the same sign, no matter what are the nonexceptional points x and z of this space. On denoting det J(x) I d(x) by dV(x) it follows that

dV(z) = dV(x) It is easy to we that under a transformation of parameters, x - x', whose Jacobian matrix x: is nonsingular at x', dV(x) is multiplied by a nonzero

30


constant. Indeed, J(x) = J(x, y) = y. - y'., = yv'yxX:' so that dV(x) -* I det y' II det J(x) II dot x=' II d(x') = f Idet y'y I dV(x) the positive or negative sign being used according as det x.- is positive or negative. Restricting ourselves to those changes of parameters for which det xs- is positive, we see that the only effect upon dV(x) of such a change of parameters is its multiplication by a positive constant. Such a multipli-

cation is of no significance for us and we term I dot J(x) I d(x), or its product by any positive constant, the element of volume of the parameter group. We express, then, as follows the content of the relation dV(z) _ dV (x) :

The element of volume of a parameter group is insensitive to all left-tramp latioms of the group.

The undetermined multiplying constant may be fixed by requiring that the volume of the group, i.e., the volume of the parametric space when this

space is assigned the element of volume dV(x), be unity. When we set dV(x) = I det J(x) I d(x), the volume of the group is V = fI det J(x) I d(x) the integration being over the entire parametric space (the exceptional points of the parametric space not contributing, by virtue of our hypothesis concerning these exceptional points, anything to V). Since the parametric space is, by hypothesis, bounded and closed, and since J(x) is continuous over this space, V is finite and we may take (1/V)JI dot J(x) I d(x) as our normalized element of volume, i.e., the element of volume which makes the volume of the group = 1. Now, let #(x) be any function which is properly integrable, and, hence, bounded, over the parametric space of our parameter group. We multiply o(x) by I det J(x) I and integrate this product over the parametric space. We term the integral of this product, rather than the integral of O(x) itself,

over the parametric space, the integral of #(x) over the group, and we denote this integral by I(#) ; thus

I(4,) = f c(x)I det J(x)I d(x) = fcb(z) dV(x) On division by V or, equivalently, on replacing I dot J(x) I d(x) by the normalized element of volume, we obtain what is known as the average, Av ¢, of ¢(x), over the group:

Av ¢ = V f(x)JdetJ(x)Id(x);

V=

f I dot J(x)I d(x)

The reason for using dot J(x) I d(x) rather than d(x) as the element of volume of the group is the following. Let x --I' z = cx be any left-translation of the parametric space and consider the function, p(x), of x which


31

is defined as follows:

¢(x) ='A(z) = ¢(cx) Then we have the following striking result: The integral of 4'(x) over the group is, no matter what is the nonexceptional point c of the parametric space, the same as the integral of ¢(x) over the group.

The various functions +'(x) obtained in the way just described from the given function O(x), by varying the nonexceptional point c of the parametric space, are known as left-translations of O(x), and we may rephrase the result just stated as follows: All Wt-translations of O(x) have the same integral over the group as does

4,(x) To prove this very useful result we have merely to observe that

I NI) = f 4'(x) dV(x) = f -A(z)dV (x) = f ,(z)dV(z) = 1(.p) The, equality jf (z)dV (x) = f c(z)dV (z) is an immediate consequence of the rule for changing variables in a multiple integral; thus, on changing the variables of integration, in the integral f (z)dV(x), from x to z we obtain the relation

f t(z)dV(x) = f ¢(z)( dot J(x)II det x, ! d(z) and, since x: = J-'(x) J(z), I dot x, ( = I det J(x) -' ( dot J(z) , so that the right-hand side of the equation just written is

f O(z)l det J(z)Jd(z) = f cp(z)dV(z) EXAMPLE

For the 1-dimensional unitary group, or, equivalently, the 2-dimensional rotation group, each of which is a 1-parameter group, the calculation of the element of volume of the group is trivial. For the unitary group, X = exp ix,

and for the rotation group X = i cs

s I,

c

c = cos x, s = sin x,

-r < x < ir. In either case the left-translation of the parametric space which is induced by the left-translation X -> Y = AX of the group is x -),

y = a + x, the addition being understood to be module 2r, so that -r < a + x < r. Then y= is the 1-dimensional identity matrix, no matter what base-point we choose, and dV(x) is simply dx, so that I(4) = j_r ¢(x)dx. The various left-translations of #(x) are the functions O(c + x),

89


-ir < c < r, and the insensitiveness of integration over the group to lefttranslations is expressed by the formula

J'.O(e + x) dx =

! ¢(x) dx

The normalized element of volume of the 1-dimensional unitary group, or of the 2-dimensional rotation group, is (1/2ir)dx.

6

The Characteristic Matrices of a Matrix Group

The n-dimensional unitary and rotation groups are not only parameter groups but also matrix groups-i.e., each element of each of them is a nonsingular n-dimensional matrix. Denoting the typical element of an n-dimensional matrix group by X and the corresponding point of the parametric space by x the elements of the N X 1 real matrix f are the real and imaginary parts of the elements of X, arranged in any order, and the statement that the r column-matrices of the N X r real matrix fs are linearly independent is equivalent to the statement that the r matrices X.1 , - , X., are linearly independent in the real field. We shall find it more convenient to deal with the matrices X-'X:i , , X-'X., , and we denote by 8X the n-dimensional matrix

8X = X-'X,tdx' +

+ X-'X.,dx'

where dx', , dx' are arbitrary real numbers. In this notation, then, we have the following characterization of a nonexceptional point of the parametric space of our n-dimensional matrix group: A point x of the parametric space is a nonexceptional point, for the system of parameters x, if, and only if, the n-dimensional matrix 8X, evaluated at x, is different from zero no matter what is the nonzero real matrix dx.

The base-point y of our parametric space is, by hypothesis, nonexcep-

tional, and so the r n-dimensional matrices Y-'Y i ,

,

Y-'Y,, are

linearly independent in the real field.

For the n-dimensional unitary group we have X-' = X* and 6X= E,_ X'X1,dxj. On differentiating with respect to xj, j = 1, ... , ns, 38

84


the relation X*X = B. we obtain XsiX + X*X=r = 0 so that the star of X*Xse is its negative, which implies that (8X) * = -OX. Thus SX is of the form iH, where H is an n-dimensional Hermitian matrix, and we say that OX is anti-Hermitian. The linear vector space whose elements are arbitrary anti-Hermitian n-dimensional matrices, it being understood that the coefficients of the linear combinations involved lie in the real field, is n2-di-

mensional, a basis being furnished by the following n2 anti-Hermitiun matrices: , n, is the diagonal n-dimensional matrix all of whore (1) N, , p = 1, diagonal elements are zero save the pth, which is i. (2) Al,, , p < q, is the n-dimensional matrix all of whose elements are zero save the element in the pth column and 9th row, which is 1, and the element in the pth row and 9th column, which is -1.

(3) N,,, p < q, is the n-dimensional matrix all of whose elements are zero save the elements in the pth column and qth row, and in the pth row and qth column, which are each = i. We term the n2 matrices N,, M,,, N,Q the characteristic matrices of the n-dimensional unitary group, and we denote by Al the 1 X n2 matrix (N1 , - - - , Na , M12, - , Ma-1, a , N12 , , Na-1, ) whose elements are these characteristic matrices arranged in the order indicated. In this notation the fact that then' characteristic matrices (N, , M,Q , N,Q) constitute a basis for the linear vector space whose elements are arbitrary antiHermitian matrices may be expressed as follows: If M' is any 1 X n' matrix whose elements are anti-Hermitian matrices, then M' = 11fF where F is a real a2-dimensional matrix which is nonsingular if, and only if, the elements of Al' are linearly independent in the real field. In particular, we may take M' to be the 1 X n2 matrix whose elements , n2, where x is any nonexceptional point of the are X*X,i , j = 1, parametric space; when we do this F is a nonsingular matrix function of x.

Let us now consider an arbitrary left-translation, X -> Z = CX, of the n-dimensional unitary group. Upon differentiating the relation Z = CX , n2, we obtain the relation Eiti1Z=+zss = with respect to x1, j = 1, CX=r = ZX*Xs, or, equivalently, k-l

Z*Z 4, =X*X1,

The left-hand side of this equation involves the two arbitrary points, x and z, of the parametric space, while the right-hand side involves only x. Upon evaluation of the left-hand side at z = y, when- y is the freely chosen base-point of the parametric space, we obtain the relation as

kl Mk

=X X;,

35


where AIk' = Y*Ydk . This relation may be more conveniently written in the form

M'J(x)dx = SX dx being an arbitrary n2 X 1 real matrix and, since AI' = A!F(y), we have

aX = AIF(y)J(x)dx If, then, we succeed in expressing 8X as a linear combination of the characteristic matrices 31i , j = 1, , n2, the coefficients of this linear combi-

nation will furnish us with n2 real linear forms in the elements of the arbitrary n2 X I matrix dx and the matrix of these linear forms will be F(y).I(x). Thus the product of d(x) by the absolute value of the determinant of this matrix serves as the element of volume of the n-dimensional unitary group. Of the n2 characteristic matrices of the n-dimensional unitary group, , N. and the n(n - 1)/2 n(n + 1)/2, consisting of then matrices N, , matrices Npq , p < q, are purely imaginary symmetric matrices, while the

remaining n(n - 1) /2, the M, , p < q, are alternating real matrices. For the n-dimensional rotation group, the n(n + 1)/2 purely imaginary characteristic matrices do not appear, the characteristic matrices of the n-dimensional rotation group being all alternating real matrices. If Mj and

AIk are any two of the n2 characteristic matrices of the n-dimensional

unitary group, so that AI j* = - M j

(Alit) * = AIk*A1 j* =

ill kAf j .

,

Mk* _ -Mk, we have

Hence (MjMk - MkMj) * =

- (M j2lk - MkM j) . AI jAfk - AIkM j is termed the commutator of AI j and AIk , in that order, and is denoted by the symbol (MjMk) ; thus (M jAtk) is, like AI j and AIk , anti-Hermitian, and we may write

(MjMk) = Mcjk where c jk is a real n2 X 1 matrix. The various relations obtained in this way by assigning different numerical values to j and k are known as the commuting relations of the n-dimensional unitary group; in these relations we may assume that j < k since (AIkMj) = - (MjMk), so that ckj = -cjk . The subset of these commuting relations obtained by restricting ourselves to the real characteristic matrices AI, , i.e., to the characteristic matrices of the n-dimensional rotation group, are the commuting relations of this latter group. EXAMPLE

The 3-dimensional rotation group. Here M = (M, , M2 , Ills), where 0

-1

M,=16112= 1

0

0 0

Ms=M13= 0

0 0

0

0

0

1

0

0

-1

0,

0

0 0 0

0

1

Ma=M2= 0

0

-1

0

W


and a simple calculation yields

(M1M2) = MIM3 - MSMI = M3 ;

(MIM3) = -M2 ;

(M2M3) = M1

It is usual in theoretical physics to introduce the basis for the linear vector space, whose elements are alternating 3-dimensional real matrices, which is furnished by the formulas

M2'=-M3=-Ma,

Ml'=Ma=M23,

Ma'=MI=M12

In this basis the commuting relations for the 3-dimensional rotation group appear in the form (MI'MY') = M3, (Ma'nta') = MI', (M3'MI') = Ms', However, this symmetrization of the commuting relations is peculiar to the value 3 of n. Since the logarithmic derivative of det X with respect to xf is the trace

of XX.i , we

obtain the characteristic matrices of the n-dimensional unimodular unitary group by restricting ourselves to those anti-Hermitian matrices whose trace is zero. For example, while the four characteristic matrices of the 2-dimensional unitary group are

MI-10

0,

M3=10

01,

M3=10

-111, 0

i

i, 01

the three characteristic matrices of the 2-dimensional unimodular unitary group are

M,

=1i 0

-01,

i

M10 s= 1

-11

0'

Ma

i

10

iI 0

Note that M1 = io,, Ma = -ioy, Ma = -io., where oz , o and o, are the Pauli matrices. The commuting relations for the 2-dimensional unimodular unitary group are

(M,M2) = -2M3,

(MIM3) = 2M2,

(M32113) = -2MI

If we change the basis for the linear vector space whose elements are antiHermitian 2-dimensional matrices with trace zero, by multiplying each of the three characteristic matrices M1, M2, 1113 by j and changing the sign of M2, we obtain the same commuting relations as for the 3-dimensional rotation group. This is evidence of a strong connection between these two groups, the 3-dimensional rotation group and the 2-dimensional unimodular unitary group. This connection is of fundamental importance in theoretical physics and we shall examine it closer in a later chapter.


37

As an example of the application of the characteristic matrices of a matrix parameter group to the calculation of the element of volume of the group, we now calculate the element of volume of the 2-dimensional unimodular unitary group. The typical element of this group is of the form U = exp ip 0

-s exp -iv

0

c = cos di

c s = sin exp -ii4 s exp C iv where 0 is a longitude angle and or and 0 are latitude angles. A simple

calculation yields the relations

-exp -iv

0

U*U, = exp iv Z

= ics exp iv U:Us I i(e$ - 82) = -2ics exp iv

U* U.

sin vMs

cos uMs

I

ics exp _ag zv I

I

= asMl - cs sin vMs + ca cos uMs

-tics exp -io -i(c2 - s2)

= cos 24,M, + sin 20 sin uMg - sin 20 cos

GA13

Thus SU = M,(s'de + cos 200) + M2(cos vd¢ - cs sin ado' + sin 20 sin add)

+ llfs(sin ado + cs cos ado' - sin 20 cos vd$)

and we have to evaluate the absolute value of the determinant of the 3-dimensional matrix

cos 2

0 cos a

-cs sin a

sin a

cs cos a

ss

Not a -sin 24, cos v sin

Adding twice the second column to the third, we obtain the matrix 0

82

1

cos or

- cs sin v

0,

sin or

cs cos or

01

whose determinant is cs = I sin 20. Thus I sin 20 d(¢,, v, P), or the product

of this by any positive constant, is the element of volume of the 2-dimensional unimodular unitary group. EXEncIsE 1

Show that I sin 2¢ d(4, a, 0, , 02), or the product of this by any positive constant, is the element of volume of the 2-dimensional unitary group.

38


EXERCISE 2

Show that the normalized element of volume of the 2-dimensional unimodular unitary group is (1/4,2) I sin 221 d(#, o, 0), and that the normalized element of volume of the 2-dimensional unitary group is (1 /&r2) I sin 20 1 d(o, o, of ,102).

The element of volume of the 3-dimensional rotation group is easily obtained in the same way. Using the Eulerian parameters we have R = R12(O)R1a(0)R12(1P) and

R*R# = 11112, R*Re = R') n'f12R12(r#) = cos OV12 - sin ¢df23 R*Re = R s(14)Ru(B)11112R1a(0)R12(¢)

= cos 011f12 - sin 0 sin #M13 - sin 0 cos p f23

We have, then, to evaluate the determinant of the 3-dimensional matrix 0

cos 0

cos >G

-sin 0 sin ' -sin 0 cos ¢

1

0 0

-sin y4

Thus the desired element of volume is sin 0 d(#, 0, 0), 0 lying in the interval 0 < 0 < r, or the product of this by any positive constant. The normalized element of volume is (1/87") sin 0 d(#, 0, -6), 0 and.0 being longitude angles. EXERCISE 3

Determine the element of volume of the 3-dimensional rotation group in terms of the modified Eulerian parameters furnished by the factorization R = Rn3(l2)Ru(0)R13(41). (Answer: cos 0 d(¢1 2), 0 lying in the

interval - (7r/2) < 0:5 it/2.) EXERCISE 4

-, the

Show that, on defining 11fp, , p > q, by the relation .11p, _ Al,, commuting relations for the n-dimensional rotation group are as follows:

(1) (11fp,M,.) = 0 if the four numbers p, q, r, s are all different (2) (Mp,111p,) = IV,., the three numbers p, q and s being all different. EXERCISE 5

Show that, on defining Np, , p > q, by the relation Np, = N,,, the commuting relations for the n-dimensional unitary group are given by the relations of Exercise 4 and the following:


39

(1) (N,Nk) 0(2) (NJAlp,) = 0 if the three numbers j, p and q are all different.

(3) (Npolp,) _ -N,,

.

(4) (N,Np,) = 0 if the three numbers j, p and q are all different. (5) (NpNpe) (6)

Alp,

.

0 if the four numbers p, q, r, s are all different. (7) (1Ip,Np,) = N,, if the three numbers p, q and s are all different.

(8) (2Ip,Np,) = 2(N, -N,), the two numbers p and q being different. 0 if the four numbers, p, q, r, s are all different. (10) (Np,Np,) _ ill,, , the three numbers p, q and a being all different. (9)

7 The Adjoint Representation of the n-Dimensional

Unitary Group

We may repeat for right-translations, X -- Z' = XC, of the n-dimensional unitary group the argument which led, when made for left-translations,

X -4 Z = CX, to the concept of the element of volume, dV(x) = I det J(z) I d(x), of the group. When we do this we obtain a new element of volume, d'V(x) _ I det J'(x) I d(x), where J'(x) is the Jacobian matrix zs, evaluated at any convenient base-point y', which is any nonexceptional point of the parametric space in the system of parameters we have adopted. We shall take as our base-point for right-translations the same point that

we chose as our base-point for left-translations, so that y' = y. If ¢(x) is any function which is integrable over the parametric space of the n-dimensional unitary group the various functions, /(x), which are obtained from the formula

V(x) = 4,(z') _ 4,(xc) where c is any fixed nonexceptional point of the parametric space, have the same integral over the parametric space when we use d'V(x), rather than d(x), as the element of volume of the parametric space. Thus, if we define I'(#) by the formula

I'(4) = f 4,(x) d'V(x) 40


41

the relation

I' W) = I'(0) is valid no matter what is the nonexceptional point c of the parametric space. Thus we have, apparently, two elements of volume, dV(x) and d'V(x), of the n-dimensional unitary group, the first of which is adapted to left-translations, and the second to right-translations, of the group. It is,

however, a fact that det J'(z) = dot J(x) so that the two elements of volume, dV(x) and d'V(x), are the same. Once we have proved the equality

of dV(x) and d'V(x) we shall know that dV(x) is adapted to right-translations, as well as to left-translations, in the following sense: If 4(x) is any function which is integrable over the parametric space of the n-dimensional

unitary group and we set '(z) = 4(cx), ¢'(x) = 4(xc), where c is any nonexceptional point of the parametric space, in the system of parameters x, then

I(#) = I(o) = I(+!-'),

where

I(4) =

J

4(x) dV(x)

this relation being valid no matter what is the nonexceptional point c of the parametric space.

In order to prove that det J'(x) = dot J(x), we follow the left-translation x -* z = ay 'x of the parametric space of the n-dimensional unitary

group by the right-translation z - to = za 'y of this parametric space, obtaining in this way the transformation x --> to = a#- 'xa 'y, to -+ x ya 'wy 'a, of the parametric space into itself. It is clear that when x = y so also does to, no matter what the nonexceptional point a of the parametric space. Upon differentiating the relation W = A Y*XA*Y with respect to x, AY*XA*Y, which yields, on evaluation we obtain the relation

at x = y (since to = y when x = y) the relation Yg(w=)., = or, equivalently,

Y*AY*YyA*Y

The 1 X n2 matrix, M', whose elements are the n-dimensional matrices Y*Y , , j = 1, - - , n2, is of the form MC, where C is a nonsingular n2-dimensional real matrix and M is the I X n2 matrix whose elements are the characteristic matrices of the n-dimensional unitary group. Using the re-

lation w: = wzs, we we that (ws).-y = J'(a)J-'(a); indeed, when x = y we have to = y and z = a and it follows, since to = za'y, that (w,), J'(a) and, since z = ay x, (zs),, = J(y, a) = J-'(a). Thus the statement that dot J'(a) = (let J(a), no matter what the nonexceptional point a of the parametric space, is equivalent to the statement that the n2-dimensional

4S


matrix, (w3), is unimodular no matter what the nonexceptional point a of the parametric space, and we proceed to prove this statement.

It follows from the relation M'(w.), = Y*AAPA*Y that the n2-dimensional matrix (w:) is a function of the point 'a of the parametric space and we denote it, accordingly, by D(y 'a), so that

llf'D(y'a) = Y*AM'A*Y On replacing a by by-'a, where b is any nonexceptional point of the parametric space, we obtain the relation

llf'D(y'by'a) = Y*BY*AM'A*I'B*Y = Y*BM'D(y'a)B*Y , n2, is Now, the qth element of M''D(y'a), q = 1, (D(y 'a)j4P and so the qth element of Y*BM'D(y'a) B*Y is Ev_, (Y*BMp B*Yj (D(y'a) j QP Since

Y*Bhfp B*Y = E:=,M.'{D(y'b) j P*

it follows that the qth element of Af'D(y'by'a), namely

=01'.1 D(yby'a) j is

E: 1E; .1M.'1 D(y'b) j r (D(y 'a) j QP

Since the n2 matrices M,', are linearly independent, by virtue of , the fact that our base-point y is a nonexceptional point of the parametric space, we have the relation

(D(y'by'a)je' = Ep-i(D(y'b)j$ (D(y 'a)j,P so that

D(y'by'a) = D(y'b)D(y'ai On writing, for a moment, y 'a = a', y 'b = b' we have the relation D(b'a') = D(b')D(a'), where a' and b' are any two nonexceptional points of the parametric space. We know, from the relation D(y'a) =

J'(a)T'(a), that the matrices D(a') are all nonsingular; furthermore, when a = y, D(y 'a) is the n2-dimensional identity matrix, by virtue of the relation M'D(y'a) = Y*AM'A*Y, and so the matrices D(a) constitute a group. The relation D(b'a') = D(b')D(a') tells us that the product of any two elements of this matrix group corresponds, in the correspondence


43

a -+ D(a), to the product of the corresponding elements of the n-dimensional unitary group, and we term the collection of n2-dimensional matrices D(a) an n2-dimensional representation of the n-dimensional unitary group. The n-dimensional unitary group possesses many other representations, of varying dimensions, and the particular representation, of dimension n2, which we have here encountered is known as the adjoint representation of the n-dimensional unitary group.

The n2-dimensional representation a -- D(a) of the n-dimensional unitary group furnishes us, since det D(ab) = det D(a) det D(b), the 1-dimensional representation a --> det D(a) of this group. If p is any integer, positive or negative, it follows that, in this representation, a" - {det D(a) } m

so that the representation is not bounded over the group unless det D(a), which is a real number, is 1 or -1. Since det D(a) is a continuous function of a, and since the parametric space is bounded and closed, dot D(a) is bounded over the parametric space, and we know that det D(a) is either 1 or -1, no matter what is the nonexceptional point a of the parametric space. This implies, by virtue of the continuity of det D(a) and the fact that every exceptional point has, by hypothesis, nonexceptional points

arbitrarily close to it, that det D(a) is either I or -1, no matter what is the point a, exceptional or not, of the parametric space. When a is the identity point e of the parametric space, D(a) is the n2-dimensional identity

matrix so that det D(e) = 1. Hence, since the parametric space is connected, det D(a) = 1, no matter what the point a of the parametric space. Thus we have the following result: The adjoint representation of the n-dimensional unitary group is unimodular

and this implies that det J'(x) = det J(x), so that d'V(x) = dV(x). (Note 1. The argument given above shows that every continuous real representation of !the n-dimensional unitary group is unimodular and that the determinants 'of the matrices of any continuous complex representation of the n-dimensional unitary group are all complex numbers of modulus unity. It is a noteworthy fact that mere qualitative knowledge-namely,

the continuity of the representation and the boundedness, closure and connectedness of the parametric space-furnishes a quantitative result: namely, the precise value of the determinant of any matrix of the representation, when this representation is real, and the precise value of the modulus of this determinant when the representation is complex.)

(Note 2. The argument we have given above for the n-dimensional unitary group is applicable to the n-dimensional rotation group. For this group the adjoint representation is of dimension n(n - 1)/2. For example, the adjoint representation of the 3-dimensional rotation group is 3-dimensional, while the adjoint representation of the 3-dimensional unitary group is 9-dimensional.)

44


(Note 3. We may also apply the argument given above to the n-dimensional unimodular unitary group. For this group the adjoint representation is of dimension a2 - 1, the traces of the various characteristic matrices being all zero. For example, the adjoint representation of the 2-dimensional unimodular unitary group is 3-dimensional.)

The fact that the adjoint representations of the n-dimensional unitary group and of the n-dimensional rotation group are unimodular greatly facilitates the calculation of the elements of volume of these groups. We know that if a' = V'a is any nonexceptional point of the parametric space, then

M'D(a') = A'M'A'* and we may write this relation, on dropping the primes from a' and A', in the form

M'D(a) = AM'A* Furthermore, SX = M'J(x)dx, so that AOXA* = M'D(a)J(x)dr Since det D(a) J(x) = det J(z) we see that, for the n-dimensional unitary group, instead of expressing SX as a linear combination of the matrices M1', ... , X., or of the characteristic matrices M1, . . , M.s, we may just as well express A8XA* as a linear combination of these matrices, A being any n-dimensional unitary matrix which may or may not vary with x.

Similarly, for the n-dimensional rotation group, we need only express REXR*, where R is any n-dimensional rotation matrix, which may vary with X, as a linear combination of the characteristic matrices of the group. For the rotation group the calculations are simplified by leaving the real

field; we write RSXR* in the form M"K(x)dx, where the n(n - 1)/2 constitute a basis, in the complex field, for the matrices M'1', MS, linear vector space whose elements are alternating n-dimensional matrices, not necessarily real. In other words, M" = MC, where C is a nonsingular

(n(n - 1)/2)-dimensional matrix whose elements are complex numbers.

Then RSXR* = MCK(z)dx so that I det CK(x) I d(x) serves as the element of volume of the n-dimensional rotation group; since this element

of volume is indeterminate to the extent of a positive multiplicative constant, it follows that I det K(x) I d(z) also serves as the element of volume of the n-dimensional rotation group. EXAMPLE 1

The element of volume of the 3-dimensional rotation group in terms of the Eulerian parameters'', 0, 4.


46

Here X = R,3(4)Ra(9)R,2(#), so that SX = M,sd j - R1a(w)Mla R,2(#)do + R s(#)R,a(O)M12R a(O)R,2(#)do. Hence R,z(#)aXR, (#) = M,sd¢ - 11I,3dO + R,a(O) M12Ria (O) d4 = M,$(41 + cos 0 dd) - M,3d9 - lll2, sin W o. The absolute value of the determinant of the 3-dimensional matrix 1

0 0

0

Cos O

0

-sin O

-1

0

is sin 0, and so sin Od(#, 0, 0) serves as the element of volume of the group. EXAMPLE 2

The element of volume of the 3-dimensional rotation group in terms of the class and in-class parameters. Any element, X, of the 3-dimensional rotation group may be written in the form Ri (t)Ri (O)Rsa(a)Ru(O)R,a(cb), where a is the class parameter

and 0, f are the in-class parameters. On denoting R,2(0)R,a(i) by R we

have the relation XR* = R*Rsa(a), so that (dX)R* + XdR* _ (dR*)RR,(a) + R*dRaa(a). Upon multiplying this equation, on the left,

the relation (SX)R* + dR* R*Rsa(a) (&R*)Rsa(a) + R*BRn(a) or, equivalently, R(SX)R* Rz*s(a) (OR*)Rs,(a) - SR* + SRsa(a). Now Rsa(a)M,2R$a(a) _ Cos a4f,s - sin aM,a, Rsa(a)M,aRn(a) = sin alll,2 + Cos aM,a, so that

by X* = R*Ri (a)R, we obtain

Ra*a(a)M' Rsa(a) _ (1/z)M2" Rza(a)M'Rs(a) = zMl ; where M'I' = M12 + i3fla , Ms" = Mls - iM,a and z = exp ia. We take Ms" and Ma = Msa as our basis, in the complex field, for the linear vector space whose elements are alternating 3-dimensional matrices, and we express R (i Y) R* as a linear combination of 11l i , M2", M3". Since R* = Ra(.O)Ri(0) we have the relation 111; ,

6R* = R,s(0)6Ra(b)R

s(e) + aR *(0) 13

_ -R12(0)11la3R s(O)d0 -

(cos OM13 + sin OMn)do - M,adO

_ -4Bli (d@ - i cos Odd) - ?Als (d0 + i cos Od4i) - Ma" sin Od¢ Hence Rsa(a) (SR*)Rsa(a)

= - (z/2) M; (dO - i cos OdO) - (1 /2z) Ms (do - cos Od4) - Ma" sin Odd

so that

46


R(SX)R* - 1

2

z Mi (do - i cos Odd)

+

11 z

2

Ms (d0 -I- i cos Odd) + Ma da

Hence dV(x) is the product of d(a, 0, ¢) by the modulus of the determinant

of the 3-dimensional matrix whose first row-matrix is ((1 - z)/2) (0, 1, - i cos 0), whose second row-matrix is ([1 -(1/z)]/2) (0, 1, i cos O), and whose third row-matrix is (1, 0, 0), or the product of this modulus by any positive constant. Since (1- z) (1 - (11z)) = 2(1 - cos a) = 4 sins a/2, we may set dV(x) = sins (a/2) cos 0 d(a, 0, 0) - sins (a,'2) da cos Od(0,

Thus the element of volume of the 3-dimensional rotation group may he written as the product of two factors, the first of which, d V (a) = sins (a/2) da,

involves only the class parameter, and the second of which, dV (0, 0) = cos0 d(0, ¢) involves only the in-class parameters. If we are dealing only with class functions we require only the class factor of the element of volume of the group in order to obtain the average over the group of any

such class function, f = f(a), as this average is simply the quotient of f f (a) dV (a) by f dV (a) . Since fo'sin$ (a/2)da = r/2, the normalized class

factor of the element of volume of the 3-dimensional rotation group is (2/w) sins (a/2) da and, since for dcb f''(,,2) cos Od0 = 4-r, the normalized inclass factor of the element of volume of the 3-dimensional rotation group is (1/4w) cos Od(0,#). The complete normalized element of volume is (1/272) sins (a/2) cos Od(a, 0, ¢), the parametric space being furnished by the rela-

tions 0 < a < r, - (ar/2) < 0 < ir/2, -ir :5

< r.

8

The Class Factor of the Element of Volume of the n-Dimensional

Unitary and Rotation Groups

We know that any n-dimensional unitary matrix, A, may be written in the form UD(a) U*, where D(a) is the n-dimensional diagonal matrix whose diagonal elements are the characteristic numbers

z, = exp ial ,

,

z = exp ia

of A, arranged in any order, and U is an n-dimensional unitary matrix which involves only the in-class parameters of A. Furthermore, SU is a linear combination of the n2 - n characteristic matrices M,g , N,g , p < q, of the n-dimensional unitary group, and the coefficients of this linear combination do not involve the class parameters a,, - - - , ap of A. On setting Al yg = Mpg + iN,g , NPe = Alps - iN,,, the n2 matrices, N, , - , N. , Al'. , pg , p < q, serve as a basis for the linear vector space whose elements are anti-Hermitian n-dimensional matrices, the coefficients of

yN

the linear combinations involved being such that the coefficient of Nn,, is the conjugate complex of the coefficient of MP",. All the elements of MD4 are zero save the element in the pth row and qth column, and so

D*(a)MrgD(a) = zpz,Mrc Similarly, since all the elements of N;g are zero save the element in the pth column and qth row, D*(a)N,,D(a) = zgx,Npg . On differentiating the

relation AU = UD(a) we obtain the equation (dA) U + AdU = 47

48


(dU)D(a) + UdD(a) and this equation yields, on multiplication, on the left, by A* = UD*(a) U*, the relation (6A) U + dU = UD*(a) (WU)D(a) + U`SD(a)l Hence

U*(SA) U = D*(a) (bU)D(a) - aU + 6D(a)

Writing SU = Ep 0, and the range of integration if p = 0. Hence, only the k! terms of our double summation for which (q) = (p) contribute anything to the integral

f(C(k - 1, - - , 1, 0))s d(a), and each of these terms contributes the same amount, namely (2r)", since all but one of the ranges of integration are 0 < a C 7, the remaining one being -7r < a < s. Thus the normalized class factor of the element of volume of the 2k-dimensional rotation group is furnished by the formula

dV(a) =

k! (27r)A;

(C(k - 1,

, 1,

12 d(a)

For example, the normalized class factor of the element of volume of the 4-dimensional rotation group, for which k = 2, is dV(ai, a2) = 2L (cos ai - cos as)2 dal dai


63

The standard representative of any class of the (2k -i- I)-dimensional rotation group is D(a) = R22(ai) R2k.2k+,(ak), where all of the class parameters as, - , ak range over the interval 0 < a < r. Any (2k + 1)dimensional rotation matrix, A, may be written in the form RD(a)R*, where the (2k + 1)-dimensional rotation matrix R involves only the 2k2 in-class parameters of A. We adopt as a basis for the linear vector space whose elements are real alternating (2k + 1)-dimensional matrices, in addition to the matrices M2s , , M2k.2k+1 , Mgi,2j , Msk,2j+1 , M2i+1,2j llfik+1.2j+1, which we encountered when studying the 2k-dimensional rotation group, the matrices M2"i = M1.21 + 1M1.21+1 ,

11121+1 = 11f1,2i - iM12i+1,

i

k.

(Note. The increase, by unity, of the various labels attached to the matrices

encountered in the study of the 2k-dimensional rotation group is due to the fact that D(a) is now R22(ai) Rsk,sk+1(ak) rather than R2k-1.sk(ak) as it was before.)

R12(a1)

Since Rsi.2:+1(ai)M2iR2k,2k+i(ai) = zllf2"i and since R2j,2j+l(aj) commutes with M2i , if j ; i, we have the relation D*(a) Bf2iD(a) = ziM2i and this relation implies its conjugate, namely, D*(a)M2i+1D(a) = *illf2i+l . We see, then, on repeating the argument used in the case of the 2k-dimen-

sional rotation group, that the class factor of the element of volume of the (2k + 1)-dimensional rotation group may be taken to be furnished by the formula

dV(a) _ fl (zi - 1)(zi - 1) fl (zizj - 1)(zi2j - 1) (2itj - 1)(2i2j - 1) d(a) Since

(zi-1)(i;- 1) =2(1 - cosai) = 22sin22 and since

2i sin

a; 2

v l ! cp(a;) = (zi - i)(zi + zip) = (zir+l

p = 1, 2, -

-;

n+l

-i ) - (zip-' - 2ia),

k - 1, we see, on introducing the notation s,(ai) = zij - z; = 2i sin jai

and denoting by S(k - -}, k --

9, , #, -1) the determinant of the kdimensional matrix whose qth row-matrix is

(8k_4+l(al), ... , sk-Q+l(ak) ), q = 1, ... , k

54


that we may take I S(k - #, - - , #) 1 2 d(a) as the class factor of the element of volume of the (2k + 1)-dimensional rotation group. Since f o'

I sa+;(a)l

2

dot = 21r, while JOT sp+i(a)sq+;(a) da = 0, if q Fd p, we see,

by the argument used in the case of the 2k-dimensional rotation group, that the normalized class factor of the element of volume of the (2k + 1) dimensional rotation group is furnished by the formula

dV(a) = I S(k - J, ... , 1) I2 d(al, ... , ak) k! 2a)k

For example, the normalized class factor of the element of volume of the 3-dimensional rotation group, for which k = 1, is (2/ir) sin2 (a/2) da.

9

Representations of the n-Dimensional Unitary

and Rotation Groups

Let us suppose that we can associate with each element, g, of any given group, G, a nonsingular linear transformation, T, of a d-dimensional linear vector space, the association being such that if g, -> T, , g2 -> T2 , where g, and gg are any two elements of G, then g2g, -> T2T, . Then, this being the case, we term the collection of linear transformations T a d-dimensional representation of G. We do not require that when g2 is different from g, then T2 is different from T, , but when this happens to be so we say that the representation is faithful. Every group possesses the 1-dimensional representation g -> 1, where I denotes the identity transformation of 1-dimen-

sional linear vector space, this representation not being faithful if G has more than one element. We term this representation the identity representa-

tion of G. If T is presented, in the standard basis, by a nonsingular ddimensional matrix, which we denote by D(g), the representation, g --I- T of

G is presented, in the standard basis, by the collection of d-dimensional

matrices D(g). In the basis which is furnished by any nonsingular ddimensional matrix, X, T is presented by the matrix X-'D(g)X, so that, regarded as representations of G, the collections of matrices D(g) and X-'D(g)X are not different; they are merely presentations, in different bases, of the same d-dimensional representation of G. If there exists a basis

in which all the matrices X-'D(g)X that present the representation are unitary, we'say that the representation is unitary and we proceed to prove the following remarkable theorem: 55

56


Every continuous representation of a parameter group whose parametric space is bounded and closed is unitary. In particular, every continuous representation of the n-dimensional unitary group, and of the n-dimensional rotation group, is unitary.

Our first step in the proof of this theorem is the definition of a positive Hermitian matrix, and of the positive square root of a positive Hermitian matrix. An Hermitian n-dimensional matrix, H, is said to be positive if x*Hx is positive no matter what is the nonzero n X 1 matrix x, with complex elements, and it follows at once that all the characteristic numbers of a positive Hermitian matrix are positive; indeed, if Hx = Ax, then x*Hx = Ax*x, so that A is positive since x*Hx and x x are both positive. Conversely, an Hermitian matrix H is positive if all its characteristic numbers are positive; for, if D(X) = U*HU is a diagonal presentation of H and y = U*x, then y*D(X)y = x*Hx and y*D(A)y = Ai9'y' + A9sys -I-

is positive. We denote by D(A3) the diagonal matrix whose diagonal elements are the positive square roots of the diagonal elements of D(A), i.e., of the characteristic numbers of H, and by H' the positive Hermitian matrix UD(A) U*, so that HiH' = H. Let, now, g - D(g) be the presentation, in the standard basis, of any continuous d-dimensional representation of a parameter group whose parametric space is bounded and closed. If v is any nonzero d X 1 complex matrix, D(g)v is a nonzero d X 1 com-

plex matrix, by virtue of the nonsingularity of D(g), and so v*D*(g)D(g)v > 0,

which implies that f v*D*(g)D(g)vdV(x) > 0, where x denotes the point of the parametric space which corresponds to the group element g. Writing the relation f v*D*(g) D(g) vdV (x) > 0 in the form v*Hv > 0, where H

is the Hermitian matrix f D*(g)D(g)dV(x), we see that H is positive. Observe that the existence of the integral f D*(g)D(g)dV(x) depends on the hypothesis that the parametric space is bounded and closed, in addition to the hypothesis that the representation, whose presentation in the standard basis is g -+ D(g), is continuous. If x' is any fixed nonexceptional point of the parametric space, corresponding to the element g' of G, we have

D*(9)HD(g') = f D*(g')D*(g)D(g)D(0) dV(x) = f (D(g)D(9 ))*D(g)D(g') dV(x) = f D*(gg)D(g9) dV(x) = H since integration over the group is insensitive to right-translations. Writing

H in the form H;Hi we have the relation H- D*(g')H1H;D(g')H-4 = Ed so that H;D(g')H-1 is unitary. In other words, the representation g --' D(g) is unitary.


57

Since, as we have just seen, every continuous representation of the ndimensional unitary group, or of the n-dimensional rotation group, is unitary we may, when discussing any continuous representation of either of these two groups, suppose the basis so chosen that the matrices, D(g),

which present the representation are all unitary, and we now do this. We term the d-dimensional space in which the d-dimensional matrices D(g) operate the carrier space of the representation, and we denote the representation by r. Let us suppose that r is such that there exists a proper subspace of its carrier space, i.e., a subspace of dimension > 1 and j and, similarly,

r'=,n"r,+...+m'°rq


59

where some of the m"s may be zero and we have 4 r + r' = E(mk+m'k)rk k-l

We now define what is known as the product of any two representations,

i', and r2 , of a given group G. The linear vector transformations which are the elements of r, are presented, in any convenient basis, by the for-

mula x - x' = DI(g)x, where x is an arbitrary d, X I matrix, and the linear vector transformations which are the elements of r2 are presented,

in the same basis, by the formula y -> y' = D2(g)y, where y is an arbitrary d2 X 1 matrix. We denote by x X y the ddd2 X 1 matrix whose r s element, r= 1 , , d, , 8= 1, d 2 , is xry° and we term x X y the direct product of the d2 X 1 matrix, y, by the d, X 1 matrix, x. We may write x X y compactly in the form

xXy=

x'y xd'y

and this notation suggests the extension of the concept of the direct product

to matrices of more than one column. Thus if A is any p X q matrix and B is any r X s matrix we understand by A X B the pr X qs matrix

a,'B.... aQ'B AXB= and we term A X B the direct product of B by A. If C is any q X Z matrix and D is any s X m matrix which are such that the ordinary matrix products AC and BD may be formed we have

(AXB)(CXD)=

a,'B

aa'B

c,'D

ci'D

a,9B .... a,PB WD ... c,°D

i

(AC),'BD

(AC),'BD

(AC),'BD

(AC),'BD

_

=AC XBD

In particular, when A and B are square matrices, of dimensions p and r, respectively, we have

(A X Er)(Ep X B) = A X B If x is any p X 1 complex matrix and y is any r X 1 complex matrix both x X y and y X x are pr X 1 complex matrices and

yXx=P(xXy)

60


where P is a pr-dimensional permutation matrix. For example, when p = 2 and r = 3, we have y'x

xxy=IxyI,

y$x

yXx=

I

yx

and P is the 6-dimensional permutation matrix 0 0 0

1

P=

0 0 0 0 0

0

0 0 0 0

1

0 0 0 0

1

0 0

1

0

0 0 1

0 0 0

0 0 0 0 0 1

Hence

(AXB)(xXy) = Ax X By = P-'(By X Ax) = P-'(B X A) (y X x) = P-'(B X A)P(x X y) Taking x to be the p X 1 matrix all of whose elements are zero save the jth, which is 1, and y to be the r X 1 matrix all of whose elements are zero save the kth, which is 1, x X y is the pr X 1 matrix all of whose elements are zero save the (j, k)th, which is I and the relation just written tells us

that the (j, k) th column matrix of P-'(B X A)P is the same as the (j, k) th column matrix of A X B. Varying j and k we obtain the relation

A X B = P-'(B X A)P If B is any r-dimensional matrix, E, X B is a pr-dimensional matrix

E,XB=

B 0

0 B

10

0

0 0

BI

whose determinant is (det B) '. Hence, since (A X E,) = P-'(E, X A)P, A being any p-dimensional matrix, A X E, is a pr-dimensional matrix whose determinant is (det A)' and this implies, by virtue of the relation

A X B = (A X E,) (E, X B), that A X B is a pr-dimensional matrix whose determinant is (det A) '(det B) ". In particular, when the matrices A and B are nonsingular, so also is A X B; in fact, since E, X E, is the

pr-dimensional identity matrix, A' X B-' is the reciprocal of A X B. It is easy to see that (A X B) * = (A* X B*), this being an immediate


61

consequence of the relation x* X y* = (x X y)*. Indeed, on starring the

relation (A X B) (x X y) = Ax X By we obtain the relation (x X y) *(A X B)

x*A* X y*B* = (x* X y*) (A* X B*)

= (x X y)*(A* X B*) and this implies, since the matrices x and y are arbitrary, that

(AXB)*=A*XB* It follows that if A and B are unitary, so also is A X B for, in this event,

A* = A-', B* = B-', so that (A X B) * = (A X B)-` Now let g --> A, g - B be the presentations, in any basis, of any two representations, r, and r2, of our group G. Then g -- A X B is the presentation, in this basis, of a representation, which we denote by r1 r2 , of G,

this being an immediate consequence of the nonsingularity of A X B and of the relation (A, X B1) (A2 X B2) = A,A2 X B1B2 . Since A X B = Pr-'(B x A) P, r2r, is the .;time representation of C as is r, r2 , so that

multiplication of group representations is a commutative operation. It is also associative, by which we mean that r,(r2ra) is the same as (r, r2) r3 , the parentheses being, accordingly, unnecessary; indeed, the relation (A X B) X C = A X (B X C) is an immediate consequence of the evident relation (x X y) X z = x X (y X z). For

((A X B) X C) ((x X y) X z) = (Ax X By) X Cz

=AxX (ByXCz) = (A X (B X C))(x X (y X z)) = (A X (B X C))((x X y) X z) and this implies, since the matrices x, y, and z are arbitrary, the relation

(AXB) X C = A X (BX C) If the two representations, r, and r2, of G are the same we denote rlr2 by r,2 so that if r is any representation, of dimension d, of G, r2 is a representation, of dimension d2, of G. Similarly, if the three representations,

n, r2 and r3, of G are the same we denote r, r2r2 by r,3, and so on. Thus if r is any representation, of dimension d, of G and m is any positive

integer, r' is a representation, of dimension d', of G. If r is irreducible, r" will be, if m > 1, in general reducible and one of the main problems of the theory of group representations is the analysis of r' into its irreducible constituents.

6a


If A and B are square matrices, of dimensions p and q, respectively, A X B is the square matrix al'B

a?B

...

a,'B apB

of dimension pq, so that the sum of the diagonal elements, or trace, of A X B

is the product of the trace of B by the trace of A. We indicate this result, of which we shall make much use, as follows:

Tr (A X B) = (Tr A) (Tr B)

10

The Orthogonality Relations for a Compact Parameter Group

We have seen that all continuous representations of any compact parameter group, i.e., any parameter group whose parametric space is bounded

and closed, are unitary and that this implies that, if the dimension, d, of the representation, r, is > 1, either (I) r is irreducible, i.e., there is no subspace, of dimension >_ 1 and 1 and we can show that any matrix that commutes with all the matrices which present, in any basis, the linear transformations that are the elements of r, is a scalar matrix, then we are assured that r is irreducible. For example, the self-representation, X - X, of the n-dimensional unitary group, n > 1, is irreducible. Indeed, if X is a diagonal unitary matrix all of whose diagonal elements are different, and M commutes with X, then M is diagonal. If two of the diagonal elements of M, say the pth and the qth, are different M does not commute with an n-dimensional unitary matrix whose element in the pth row and qth column is different from zero. Thus the only matrices which commute with all the matrices of the n-dimensional unitary group are scalar matrices which proves the irreducibility of the self-representation of the n-dimensional unitary group, it > 1.

The converse of the result proved in the previous paragraph is true: If r is irreducible the only matrices which commute with all the matrices which present, in any basis, the linear transformations that are the ele-

ments of r are scalar matrices. Indeed, the relation DM = MD tells us that Dm,, where mf is the jth column-matrix of Al, j = 1, , d, is a linear combination of the column-matrices of Al. Hence, by virtue of the

irreducibility of r, either (1) M is the zero d-dimensional matrix or

(2) M is a nonsingular d-dimensional matrix.

If M is a nonsingular d-dimensional matrix, it may be replaced, without

affecting the relation DDf = DID, by M' = M - XEd, where X is any complex number, and Af' is singular if A is a characteristic number of If. Hence M' is the zero d-dimensional matrix so that DI is a scalar matrix. It follows that all continuous irreducible representations of a commutative compact parameter group are of dimension 1. Indeed, D(a), where a is any point of the parametric space, commutes with all the matrices D(x) so that D(a) is scalar. Since a may be freely chosen, it follows that all the matrices D(x) are scalar, which implies, if the dimension of the representation is > 1, the reducibility of the representation. For example, all continuous irreducible representations of the 1-dimensional unitary group, or of the 2-dimensional rotation group, are 1-dimensional. If X -+ f(x)

is any continuous 1-dimensional representation of either of these two


85

groups we have f(a)f(x) = f(a -}- x), and the only continuous solutions of this equation are given by the formula f (x) = exp ikx, where k is an undetermined constant. Since f (r) = f( - r), k must be an integer. Thus the irreducible representations of the 1-dimensional unitary group, or of the

2-dimensional rotation group, are of the form

X --> exp imx, m = 0, f 1, f2, For the 1-dimensional unitary group this formula may be written as X --> z'",

in = 0, =E1, f 2,

-

where z = exp ix is the single characteristic number of the typical element

of the group; for the 2-dimensional rotation group it may be written as

Icosx sin x

cos x

0, f1, f2,

where z = exp ix and 1/z = exp (-ix) are the two characteristic numbers of the typical element of the group.

Now let X -i, D(x) be any continuous irreducible representation, of dimension d, of any compact parameter group and construct the matrix

M = fD(f')CD(x)dV(x) where C is any constant d-dimensional matrix whatsoever. Since group integration is insensitive to right.-translations we have D(a74)MID(a) = M where a is any point of the parametric space. Thus 111 commutes with all the matrices D(x), so that if is a scalar matrix no matter what the constant matrix C. Taking C to be the matrix all of whose elements are zero save the element in the pth row and qth column, which is 1, we obtain the relation

f {D(x ') }p'{D(x) )," dV(x) = k,"50' where b,' is the element in the rth row and the sth column of the d-dimensional identity matrix, and kp° is an undetermined constant. On setting

s = r and summing with respect to r we obtain, since D(x)D(x ') is the d-dimensional identity matrix, the relation Vbp° = kp° d so that

f {D(x')1a {D(x) l.° dV(x) =

V by b='

On varying the indices r, q, p, 8 we obtain what are known as the orthogonality relations for any continuous irreducible representation of any compact

parameter group On setting p = r, s = q, and summing with respect to r

68


and q we obtain the relation

fx(x1)x(x)dV(x) = 1 where x(x) is the trace, &fD(x)je°, of D(x). For the 1-dimensional , this unitary group, for which X (x) = exp imx, in = 0, :1:], t2, reduces to(1/2ir)f', (exp - imx) exp (imx) dx = 1. In general, since the representation X -+ D(x) of our compact group is unitary, x(x7) = x(x) so that f x(x)x(x) dV(x) = 1

In words: The average over the group of the squared modulus of x(x) is unity. We next consider two different irreducible continuous representations, X -, D,(x), X -> D2(x), of dimensions d, and ds , respectively, of our compact parameter group, where dz may be the same as d, . On introducing the d, X dz matrix

M = f D,(x ')CD2(x) dV(x) where C is any constant d, X d2 matrix, we see, since group integration is insensitive to right-translations, that

D,(a)M = A1D2(a) where a is any point of the parametric space. Hence, since the representa-

tion X - D, (x) is irreducible, either (1) M is the zero d, X d2 matrix, or (2) d2 >_ d, and the d2 d, X 1 column-matrices of Al are linearly independent.

On starring the relation D,(a)M = AfD2(a) we obtain the relation Dz*(a)M* = M*D,*(a)

and, since we may suppose our basis so chosen that the matrices DI(a) and D2(a) are unitary, no matter what the point a of the parametric space, this relation may be written in the form

D2(a ')M* = M*D,(a ') Hence, since the representation X -- D2(x) is irreducible, either (1') A1* is the zero de X d, matrix, or


67

(2') d, >_ d2 and the d, d2 X I column-matrices of Df" are linearly independent.

The results (1), (2), (1') and (2') may be combined as follows: either 0") Al is the zero d, X d2 matrix, or (2") d2 = d, and the d,-dimensional matrix llf is nonsingular. The alternative 2") is not acceptable, for it would imply, since D2(a) = Dr'D,(a)M, that the two representations X - DI(x) and X - D2(x) are the same. Hence M is the zero d, X d2 matrix, no matter what is the constant d, X dz matrix C. Taking C to be the matrix all of whose elements

are zero save the element in the pth row and qth column, which is 1, we obtain the relation

f {D,(x ') }r (D2(x) }, dV(x) = 0 On varying the indices r, q, p, s, we obtain what are known as the orthogonality relations for any two different continuous irreducible representations

of any compact parameter group. On setting p = r, s = q and summing with respect to r and q, we obtain the relation

f xl(x')x2(x) dV(x) = 0 or, equivalently,

f ii(x)x2(x) dV(x) = 0 It follows from these results that the analysis of any reducible continuous representation, r, of any compact parameter group into its irreducible constituents is unique. Indeed, if r, , - - , r} are the irreducible representations which occur in two hypothetical analyses of r, we may write these

two analyses as r = c' r, + .. + ciri , r = c''r, + ... + c'iri where some of the c's and some of the c"s may be zero. From the first analysis we obtain

X(x) = c'xi(x) + ... + cixi(x) and the orthogonality relations yield the relations

f xk(x)x(x) dV(x) = ?-V, so that ck

=V

f

dV(x),

k = 1,

,

j

k=1,...'j

68


Similarly, c'k = (1/V)f xk(x)x(x) dV(x) = ck, which proves that the two hypothetical analyses of r into irreducible constituents are one and the same. Furthermore,

f x(x)x(x) dV(x) =

(cr)2

+ ... + (c')2

so that, no matter what is the continuous representation r of our compact parameter group, (1/V)f x(x)x(x) dV(x) is an integer which is the sum of a number of squared integers. In particular, t is irreducible when, and

only when, (1/V)f x(x)x(x) dV(x) = 1. Since the analysis of r into its irreducible constituents is unambiguously determined by the function x(x), we term x(x) the character of the representation. When

fiThx(x)dV(x) = 1, F is irreducible and we term x(x) a simple character of our compact parameter group; otherwise, t is reducible and we term x(x) a compound character of the group. The character, x(x), of any representation, reducible or irreducible, of a compact parameter group is a class function,

since the trace of D(a4)D(x)D(a) is the same as the trace of D(m). Thus, for the n-dimensional unitary group, x(x) is a symmetric periodic function, of period 2a, of the n class parameters al , , a ; for the 2kdimensional rotation group, x(x) is a symmetric periodic function, of period 2rr, of the k-class parameters a, , , ak which is insensitive to a change of sign of any two of these parameters; and, for the (2k + 1)dimensional rotation group, x(x) is an even symmetric periodic function, of period 2w, of the k-class parameters a, , - - , ak . Any compound character, x(x), is of the form c'xk(x) + + c'xI(x) , c' are nonnegative integers and xl(x), , where the coefficients c', x;(x) are simple characters. If one or more of the coefficients c', , c', in an expression of the form c'xl(x) + . + c'x,(x) is a negative integer, this expression is not the character of any representation of our compact parameter group but it is convenient, nevertheless, to consider it, and we + c'x; (x), where the coterm any expression of the form c'x, (x) + , ci are integers, positive, negative or zero, a generalized efficients c', character of the group. Thus simple and compound characters are particular cases of generalized characters. A generalized character is a simple character if, and only if, all the coefficients c', , c' are zero, save one which is 1. The average over the group of the squared modulus of

c'xi(x) + ... + c'xi(x) is (c')2 +

+ (c')2, and if this is 1 all the coefficients c', .

,

c' are


69

zero, save one which is either 1 or -1. If the value of X(x) at the identity point, e, of the parametric space is positive the one nonzero coefficient must be 1, since Xj(e) is the dimension of the irreducible representation of our compact parameter group whose character is X j (x) . Thus we have the following useful result: A generalized character, X(x), of a compact parameter group is a simple character, i.e., the character of an irreducible representation of the group, if, and only if,

(1) The average of x(x)x(x) over the group is 1. (2) The value, X(e), of x(x) at the identity point of the parametric space is positive.

We know that any character, simple, compound or generalized, of the n-dimensional unitary group is a symmetric function of the n characteristic numbers (z, , , z,) of the typical element of the group; it follows, on

denoting the character in question by X(z), that X(z)0(z), where 0(z) is the difference product, (z1 - z2) ... zn), of the n complex numbers z,

,

-

, zp , is an alternating function of zi

,

-

, z. . If Al

,

,

X. are n nonnegative integers, arranged in nonincreasing order, whose sum is any given positive integer in, so that (X) = (Al , , A.) is a partion of m into not more than n parts, the sum Es,s f zllD, where 11=n-1+X1,

I2=n-2-1-X2, ,la=Aa

so that 11 > l2 > .. > 1 > 0, and (p) = (p, , . , pA) is any permutation of the n numbers 1, - , n, the + or - sign being used according as (p) is even or odd, is an alternating function of z, , - , z. . It is, in fact, the determinant of the n-dimensional Vandermonde matrix whose jth , n. Its quotient by &(z) is, acrow-matrix is (zl' , , za"), j = 1, cordingly, a symmetric polynomial function of degree m of (z1 , , which we denote by JA). We shall see in the next chapter that { A} is a generalized character of the n-dimensional unitary group. Anticipating this result, it is easy to see that it is a simple character of the group. In-

deed, since the class factor of the element of volume of the group is X(z) 0(z) d(a), where zj = exp iaj , j = 1, , n, the argument which showed that

f 0(z)A(z) d(a) = n!(2r)" shows that the average of the squared modulus of {A{ over the group is 1. To prove that { A} is a simple character, all that remains to be done is to show that the value of {XJ at the identity point, c, of the parametric space

is positive. Writing zj = I + e j , j = 1,

, n, and letting the e's tend to zero we see that the value of { A{ at c is the quotient of the difference

70

Lectures on Applied Mathematic8

product, A(l) = (Ii - ls) (la-, - la), by the difference product of the n numbers n - 1, , 1, 0, this latter difference product being (n - 1) !(n - 2) ! 2!. Thus {a{ is the character of an irreducible representation of the n-dimensional unitary group whose dimension is O(l)/[(n - 1) I(n - 2) ! 2!1. We shall see later that there are no other simple characters of the n-dimensional unitary group than the functions (XI or the quotients of these by a positive integral power of the determinant, det X = z, za , of the typical element of the group.

11

The Irreducible Representations of the n-Dimensional Unitary Group

Let A be any square matrix, of dimension d, and let ak' be the element in the jth row and kth column of A. Then the direct, or Kronecker, mth power of A, which we denote by A(m), is a square matrix, of dimension , d', of which the element in the (j1, th row and (k, , , k,") th akA. We denote this, for brevity, by a((k) and we say that column is ai; a((W)) is the element in the (j)th row and (k)th column of A(m). On denoting , jm) th by y the d' X I matrix of which the element in the (j) = (j) , the element in the (j)th row of A(m)y is E(k) a(k)y(k). row is y(') If (p) = (p) , - - , p,") is any permutation of the m symbols 1, - , m, , jp",) by (p) (j), and it follows from the commutativity we denote (jp of multiplication of complex numbers that a(p)(k) = auk;. Finally, we denote by (p) y the dm X 1 matrix whose (j) th element is y(')(3). Then the

element in the (j) th row of Am( (p) y) is given by the formula a((k))y(p)(k) =

A(m)((p)y)"') = (k) E

E(k)a(piikiy(p)(k)

Since, as (k) runs over its set of dm values, so also does (p) (k), it follows that the (j) th element of A(m)( (p) y) is F,(k) a(k (''y(k) which is the (p) (j)th element of A(m)y. Thus A(m)((p)y)

= (p)(A(m)y)

no matter what the permutation (p) of the m symbols 1, 71

, m.

72


If x is any d'° X 1 matrix we define its average, x+, over the symmetric group on m symbols as follows:

x+ =

1' 2 (p)x m. cr>

If (q) = (q, ,

is any permutation on the m symbols 1, , m, , (qp) runs over the symmetric group on m symbols as (p) does and so, since (q) (p)x = (qp)x, we have

(q)x+ = x+ Similarly, we define the alternating average, x-, of x over the symmetric group on m symbols by the formula

x_ =

1 E f (p)x

M! (P)

where the ambiguous sign f attached to the permutation (p) implies that we use the + sign when (p) is even and the - sign when (p) is odd. Then, by the argument just given,

f(q)x- = X_ The vectors of the carrier space of the linear vector operator which is presented by A(m), which are presented by d" X 1 matrices y which are such

that (p)y = y, no matter what the permutation (p) of the m symbols 1, , m, constitute a linear subspace of this carrier space, and it follows from the relation (p) (A(m)y) = A(")((p)y) that this subspace is invariant

under the linear vector operator which is presented by A(", no matter what the d-dimensional matrix A. We term this subspace the symmetric subspace of the carrier space of the linear vector operator which is presented by A("). Similarly, the vectors of the carrier space of the linear vector operator which is presented by A("), which are presented by d" X 1 matrices y, which are such that f (p) y = y, no matter what the permuta, in, constitute a linear subspace of this tion (p) of the m symbols 1,

carrier space, which is invariant under the linear operator presented by A(', no matter what is the d-dimensional matrix A. We term this subspace the alternating, or antisymmelric, subspace of the carrier space of the linear vector operator which is presented by A("). It is clear that these symmetric

and alternating subspaces have no vector, other than the zero vector, in common since, when (p) is odd, the two relations (p) y = y, - (p) y = y would force y to be the zero d" X 1 matrix. It is clear that if any two of the numbers j, , , j" are the same the

(j) th element of a d" X 1 matrix, y, for which d (p) y = y, no matter what is the permutation (p), is zero. Indeed, when (p) is the odd permutation which corresponds to a transposition of any two equal numbers from


73

the set j, , , j", , we have y(A)(i) = y('' and this, combined with the relation - (p) y = y, yields y(" = 0. If, then, in > d the only vector belonging to the alternating subspace is the zero vector since, when, in > d, two of the numbers j, , , jm must be equal, since each of these m indices takes on one of the d values 1, , d. We assume, accordingly, when discussing the alternating subspace of the carrier space of the linear vector

operator presented by A(), that in < d. If k, < k2 < combination of the d numbers 1, x the direct product x(k) = ek, X

< k", is any

, d, we may take as our d" X I matrix X ek. ; when we do this, (x(k)_) (" is

zero unless (j) is a permutation of (k), and when (j) is a permutation of

(k), (x(k)_)" is d l/m!, the + or - sign being used according as the permutation is even or odd, respectively. By varying the combination (k) we obtain

(d) m

(d)

= m! (dd!

m) I

matrices x(k)_ which present a set of

linearly independent vectors of the alternating subspace, and so

the dimension of the alternating subspace is at least

(d) . It cannot be

greater than (dt) since the matrix, y, which presents any vector of the alternating subspace, is of the form m!E(k) y(k)x(k)- . Thus we have the following result: The dimension of the alternating subspace of the carrier space of the linear

vector operator which is presented by A('), where m < d, is the number of dt combinations, (m) = m! (d m) , of d things, taken m at a time, and the X ek",)_, where (k) is various d' X 1 matrices x(k)- = (ek, X ek, X any combination of in of the d numbers 1, of this alternating subspace.

, d, present the vectors of a basis

For the symmetric subspace there is no requirement that m be :5d, and (k) is now any combination, with repetitions allowed, of the d numbers, in at a time. The number of such combinations is the number of different

terms in the mth power of a sum of d indeterminates. Denoting it by 4(d, m), we have 4,(d, 1) = d, %6(1, m) = 1, and

O(d, m) = 1 + O(d - 1, 1) + O(d - 1, 2) + ... + qs(d - 1, in). Thus O(d, 2) = d + q5(d - 1, 2) from which, together with the fact that ¢(1, 2) = 1, we deduce that O (d, 2) = jd(d + 1). Similarly, q5(d, 3) = (1/3I)d(d + 1) (d + 2) and, generally,

O(d, m) = MI d(d

+ 1) ... (d -l-

1) _ (ml d - 1)!

I

74


Thus the number of combinations, with repetitions allowed, of d things in at a time is the same as the number of combinations, without repetitions,

of d + in - 1 things in at a time. By the same argument as in the case of the alternating subspace, we see that: The dimension of the symmetric subspace of the carrier space of the linear vector operator which is presented by AC'°' is the number,

(d -I- m - 1)I/ml(d- 1)! of combinations, with repetitions allowed, of d things taken in at a time and where (k) the various d' X 1 matrices x(k)+ = (ek, X ek, X X is any combination, with repetitions allowed, of in of the d numbers 1, - - - , d, present the vectors of a basis of this symmetric subspace.

(Note. When in = 2, the dimension of the alternating subspace is

d(d - 1)/2 and the dimension of the symmetric subspace is d(d + 1) /2 so that the sum of these two dimensions is d2, the dimension of the carrier space of the linear vector operator which is presented by A" = A('". Thus, when in = 2, the d2 vectors obtained by combining the vectors of a basis of the alternating subspace and the vectors of a basis of the symmetric subspace constitute a basis for the entire carrier space. When in > 2 this is not the case, the sum of the dimensions of the alternating and symmetric subspaces being less than the dimension of the entire carrier spa(e.) Now let the matrices A present a representation, of dimension d, of the

n-dimensional unitary group, d >_ 2. Then, the matrices A"), in > 1, present the representation r" of this group and we know that r" is reducible, since the alternating and symmetric subspaces of its carrier space are invariant under it. We term the representation, of dimension

(d+in -1)I/ml(d- 1)I induced by 1'" in the symmetric subspace of its carrier space the symmetrized mth power of r and we denote, for the time being, this symmetrized mth power of 1' by (I''") + . Similarly, we term the representation induced by r' in the alternating subspace of its carrier space the antisymmetrized mth power of 1', and we denote, for the time being, this antisym-

metrized mth power of r by (r')-. In order to determine the characters of these representations, we may take A to be diagonal, with diagonal elements Zi , , Zd , say; indeed, if U*A U = A', then (U*) (m)A(=) Uc," = A1(m)

and the trace of A'(') is the same as that of A('). When A is diagonal, so also is A("`), the (j)th diagonal element of A('") being Z>" Za` where


m, . i = 1, - . , d, is the number of is in (j) = (ii, x(k)_, k, < k2 < < km is one of the basis vectors (ek, X A(m)x(k)_ = Zk,

(r'") - is E. =

75

, j.). If

X ek,) _ of the alternating subspace,

Zkmx(k)_ and this implies that the character of E(k) Zk, Zkm , the mth elementary symmetric function

of the d characteristic numbers of A. Indeed, if B is any nonsingular dm-di-

mensional matrix whose first ('i) ecolumn matrices are the matrices x(k)_ then B-'A(m)B has as its first (d Zk,

'

column matrices the matrices

where e(k) denotes the` d' X 1 matrix all of whose ele-

ments are zero save the (k)th which is 1. Thus: The character of the antisymmetrized mth power, (r")_ , of any representation, r, of the n-dimensional unitary group is the mth elementary symmetric function, E. = E(k) Zk, of the d characteristic numbers of the typical element of r. In particular: The character of the antisymmetrized mth power of the self-representation of the n-dimensional unitary group is the mth elementary sy)nntetric function,

am = E(k) Zk,

zkm , of the n characteristic numbers of the typical n-

dimensional unitary matrix. Zk," where, Similarly, the character of (1'm) + is H. = Fl(k) Zk, now, repetitions are allowed in the combination, (k), of m of the d numbers 1, , d. For example,

HS = E Zr2 -i- r_ X. >_ 0, of m into not more than d nonzero parts, of degree m of the characteristic numbers of the typical element of r.

In particular: The character of the symmetrized mth power of the self-representation of the n-dimensional unitary group is the complete symmetric function, hm = (k)

(

z ... .rm


76

of degree m of the n characteristic numbers of the typical n-dimnsional unitary matrix.

Since the product of the characters of any two representations of the ndimensional unitary group is the character of the product of the two representations, it follows that, if mi , - , m, is any set of positive integers, Emf and H(m) = IIm,Hm, - - Hm, are the then E(m) _ Em, Fm, characters of representations of the n-dimensional unitary group, and so any expression of the form E(m) c(m' D.) or of the form E(m) c(')H(.) ,

where the coefficients, care integers, positive, negative or zero, is a generalized character of the n-dimensional unitary group. In particular, any expression of the form Fl(n) ecm)am , or of the form Fl(n) e(m'hm , is a generalized character of the n-dimensional unitary group. We are now ready to prove that the symmetric functions, ( NJ, of the characteristic numbers of the typical element of the n-dimensional unitary

group, which we introduced in the preceding chapter, are generalized characters of the group. To do this we consider the polynomial function of degree n of an indeterminate t:

At) = (1 - zit) ... (1 -

1 - 0,1t + ast$ - ... +

Since

(1 - z,t)-1 = 1 + zit + z,zts + ... , j = 1, ... (f(t))-' = 1+hit +hi+ , II < Iz,j On denoting we have

1)'o j by o,', j = 1,

11+01'1+02'e+

, n,

t

< I z; I

-1

= 1,...n

, n, and setting o,' = 0 if j > n,

ill +hlt+h2e+

and it follows that, if m is any positive integer, the following two m-dimen-

sional unimodular triangular matrices vo

91

0

go

am-1 am-s

T. =

T,,,'

-

...

he

h1

0

he

..

10

0

...

hm-t hm-z

ho

I

where ao' = 1, he = 1, are reciprocals each of the other. If l is any positive


77

integer we denote by (hi) the I X n matrix (h,_n+i, hl-n+2 , , hi), where h; = 0 if j < 0, so that the jth row-matrix, j = 1, , n, of Tn' is (h._1). Hence (h"_1) T" is the jth row-matrix of the n-dimensional identity matrix so that z"-' zn-2

=z

(h"_,) T.

n-j

j=1, ,n

1

where z is any complex number. In other words, if 1 is any one of the n integersn - 1, , 1,0, z"-i zn-2

Z' = (hi) T. 1

z being any complex number. This relation remains valid for any positive integer l whatsoever if z is one of the n complex numbers z, , , zn . To

see this we write If(t) I-% where f(t) = (I - zit)

(1 - znt), in the

form 1 + h11 +

j, where m is any

+ hm._1tm-' + tm{hm + hm+it + positive integer. Then

f(t)thm + hm+1t + ... )t" +f(t)(1 + h11 + ... + hm-ltm-') = 1 which implies, since 1 - f (t) (1 + h11 + + hm_1tm-') is a polynomial .) is a polynomial of degree of degree 2, we have other such representations of the n-dimensional unitary group. Thus, when in = 3, we have r ® (21), whose character is

H21II - H3

in addition to 1' 0 (3) and r ® (13). We shall see in the next chapter that them are, essentially, no other continuous representations of the n-dimensional unitary group than those whose characters are furnished by the functions (XI.

12

The Analysis of any Power of a Representation of the

n-Dimensional Unitary Group

If t is any continuous representation of dimension d of the n-dimensional unitary group, we obtain from r (by restricting our attention to those of its operators that correspond to diagonal n-dimensional unitary matrices) a continuous representation of the subgroup of the n-dimensional unitary group which consists of all diagonal n-dimensional unitary matrices. This subgroup is commutative and therefore all it., irreducible representations are 1-dimensional; hence there exists a basis in which the matrices that

present those operators of t corresponding to diagonal n-dimensional unitary matrices are all diagonal. Let z, = exp ia, , - , zn = exp ia be the diagonal elements of any diagonal n-dimensional unitary matrix, X, and let Z, = exp Zi, , - - , Zd = exp i$d be the diagonal elements of the diagonal matrix which presents, when the basis is properly chosen, the operator of r corresponding to X. Then each of the P's is a continuous function of the n a's that satisfies the equation Q(ai -1 ai , ... , an -l- an') _ 0(ai , ... , an) -l- P(ai', - . , an') On setting all of the a's but one equal to zero in any one of the $'s, we obtain a function of a single variable a; in this way we obtain do functions of a , n. single variable a which w e denote b y 6ik(a) . j = l , , d, k = 1, For example, tha(a) = 2(0, 0, a, 0, - , 0). Then, the relation IBi(a,,...,an) 4-18,(at',...,a.') 80


81

implies the relation d p;(al , " , an) = p;1(al) + p,2(a2) + ... + 11,"(an), j Since p,k(a) -l- p;k(a') = 16;k(a + a'),j = 1, .. , d, k = 1, ... , n, #;k(a)

is it constant multiple of a, the multiple being an integer because each of the Z's is a periodic function, of period 27, of each of the a's. Thus 13i(ar, ..

,

a") _ Ek-1m,kak,

d; m,k an integer

which implies that Zi=zi"'...zn"'j", j= 1,...,d

It follows, since every class of the n-dimensional unitary group possesses

a diagonal representative, that, if all the integers m,k are nonnegative, the character Efel Z; of r is a polynomial function, with positive integral coefficients, of the n characteristic numbers of X; if some of the integers m,k are negative the product of the character of 1' by a sufficiently high power of det X = zl .. z" is such a polynomial function. Thus we have the following result: The product of the character of any continuous representation of the ndirncnsional unitary group by a sufficiently high power of the determinant of the group element, X, is a polynomial function, with positive integral coefficients, of the characteristic numbers of X.

Hence, the product of (det X) sx( r) by the difference product

A(Z) = (z1 - z2) ... (Zn-1 - Zn) of the characteristic numbers of X is, if the positive integer p is sufficiently large, an alternating polynomial function, with positive or negative integral coefficients, of z, , - - - , z. . A typical term of this polynomial function is of the form c(r)zl" - - zn'" where no two of the 1's are the same and the appearance of this term implies the appearance of the n! terms ±C(1)Zl'al ... zn'9n , , p,,) is any permutation of the n numbers 1, - , n, and the + or - sign is used according as this permutation is even or odd,

where (p1

respectively. We may, then, assume that l1 > I2 > > In ? 0 so that , , In = A., where the numbers A1, we may write 11 = Al + n - 1, An are nonnegative integers, arranged in nonincreasing order of magnitude.

On dividing through by o(z), it follows that (det X) "x(r) is a linear combination, with integral coefficients, of the functions (AJ which we know to be simple characters of the n-dimensional unitary group. Now X -+ det X is a 1-dimensional representation of the n-dimensional unitary

82


group, the determinant representation, and so X - (det X) " is a 1-dimen-

sional representation of the group, the pth power of the determinant representation of the group. Hence (dot X) "x(r) is the character of a representation of the group, the product of r by the pth power of the determinant representation. Denoting, for a moment, by x the character, (det X) "x(r), of this representation, the number of times that the representation contains the irreducible representation whose character is ( A) is the average of X (A) over the group; we have seen that x is of the form E(x) C().) 1) and so, by virtue of the orthogonality relations connecting the functions ( X), this average is c(A) . Hence c(,,) is a nonnegative integer and we have the following result: If r is any continuous representation of the n-dimensional unitary group the product of r by the pth power of the determinant representation of the group is, if the positive integer p is sufficiently large, a linear combination, with nonnegative integral coefficients, of the irreducible representations of the group whose characters are the functions ( A).

In other words, there are, essentially, no other continuous irreducible representations of the n-dimensional unitary group than those whose characters are furnished by the functions ( X1. EXAMPLE

X --. X, the conjugate complex of X, is the conjugate representation of

the n-dimensional unitary group. Its character is E1" i, , so that the product of this character by det X = z1 z" , is, since i, = 1/z; , pn-1 = 11 "-1)

If A"> 0, (det X)' = z1'"

z

is a factor of ;XI, the other factor

being (Al- A , A2 - A" , , A"-1 - X.1, and so there is no essential loss of generality, when considering irreducible continuous representations of the n-dimensional unitary group, in confining our attention to partitions

of any nonnegative integer m into not more than n - 1, rather than n, nonzero parts. In particular, when n = 2, all continuous representations of the 2-dimensional unitary group are furnished by the determinant representation and the irreducible representations whose characters are the functions (ml, these being the symmetrized powers of the self-representation. The symbol (0) denotes the identity representation; (11 is the self-representation; (2) is the symmetrized second power of the self-representation, and so on. The antisymmetrized mth power of the self-representation vanishes if m > 2 and the antisymmetrized second power, whose

character is (1s), is the determinant representation. When n > 2, the irreducible representation whose character is (1 ") , i.e., the antisymmetrized

nth power of the self-representation, is the determinant representation.


88

When m = 2 the carrier spaces of the symmetrized and antisymmetrized mth powers of any representation, r, of the n-dimensional unitary group exhaust the carrier space of r", but when m > 2 this is no longer the case. To find out what other representations of the n-dimensional unitary group are contained in Fm when in > 2, we first consider the product of the irreducible representation whose character is JAI by the self-representation,

z

whose character is {1) = z1 +

+ z" . One term of i(z) { A) is zill z"r"; hence, since 0(z){ A) (1) and so one term of &(z) (A} { 1) is zir1+'z2i=

is an alternating function of z1,

Ef

, z" pit contains the expression zlial+1x2''2... '.'P",

(P)

where the + or - sign is used according as the permutation

(p) _

(pi,...,p")

, n is even or odd, respectively. On dividing through by A (z) it follows that {X1 } contains { Al + 1, A2 , , A"} . Similarly, (A) 1 } contains, if A, > A2 , Al , A2 -f 1, . , A"} i if A1- A2 , 11 = 12 + 1 and E(P) f z110iz2'D3+1 . . z"1°" vanishes. In the definition of the symbol { A) we have agreed that A, > X2 >- ... > X. > 0, in which case (A) is the determinant of the n-dimensional matrix whose jth row-matrix is

of the n symbols 1,

(hij-"+1, ... , h'j)If we interchange the jth and (j + 1) st row-matrices of this n-dimensional matrix we change the sign of its determinant, and so we define {A}, when >_ A. >- 0 is violated, by one or more the ordering relation Al ? A2 >_ applications of the formula

A1,...,A;+1- 1,AJ+1,Aj+2...,A"} - -(Al,... ,AJ, AJ+i , ... ,A") In particular, when AJ+1 = AJ + 1, {A) vanishes, since, when this is the and case, both {A,, ,Aj+1-

JAI, ...,NJ,AJ+1,.A") am the same. A repeated application of this rule for taking care of disordered partitions (A) of n shows that (A) vanishes when Aj+2 = AJ + 2 or when A;.1. = A j + 3 and so on. With this convention, then, {A){1} _ JAI + 1, A2,

If (A) _ (A1 ,

,

A,)) has k < n nonzero parts, we may stop with the

84


term {Al , , Ak, 11 on the right, since the remaining terms all contain a 1 following a zero. For example,

= 11)(1) = (2) + 1121 This formula expresses the fact that the square of the self-representation of the n-dimensional unitary group is the sum of the symmetrized and (1)2

antisymmetrized squares of the self-representation. Applied to any representation, r, of the group it tells us that the square of r is the sum of the

symmetrized and antisymmetrized squares of r. We denote the symmetrized square of r by r ® (2) and the antisymmetrized square of r by r ® (12) and we write

r2=(r®(2))+(r®(1i)) To analyze r' we observe that 11)' = {11121 + 11)112) where 111121 = 131 + 1211, and {11(121 = 1211 + 11 21 + (13) = 1211 + 1131, by the rule just proved, so that 11)' = 13) +21211 + 113)

Thus the cube of the self-representation contains, in addition to the symmetrized cube, whose character is 131, and the antisymmetrized cube, whose character is 1131, the irreducible representation whose character is 12 1), this latter representation occurring twice. For any representation, r, of the n-dimensional unitary group we have

r'= (r® (3))+2(r® (2 1)) +(r®(l3)) Similarly, (1)4= 141 + 3131) + 2(22) + 3(2 121 + ,141

1116 = 151 + 4141) + 513 2) + 61312) + 512211 + 412 13+ 1151

and so on. These relations tell us that all the irreducible representations of the n-dimensional unitary group whose characters arc the functions (A) appear in the analyses of the various powers of the self-representation. The relations just written furnish the analyses

r'= (r® (4))+3(r®(31)) +2(r®(22)) + 3(r ® (212)) + (r ® (1`))

r6= (r® (5))+4(r® (41)) +5(r® (32)) +6(r® (312)) +5(r®(221))+4(r®(216))+(r®(15)) of the fourth and fifth powers of any representation of the n-dimensional unitary group; and so on. It is understood that we discard, in these formulas,


85

any {A) for which the number of nonzero parts exceeds n. For example, when n

2,

11)2 = (2) + )12) = (2) + detX(0)

{1)a = (3) + 2(21) = (3) + 2detXI1) )1)' _ (4) +31311 + 2(22) = (4) + 3 det X(2) + 2(det X)"-(0)

{1)5 = {5) +4)41) +5(32) = (5) +4detX(3) +5(detX)2(1) and so on. Although the representations r 0 (A) of the d-dimensional unitary group (where I' is any representation, of dimension d, of the ndimensional unitary group) are irreducible, the representations r 0 (X) of the n-dimensional unitary group which are induced by the irreducible representations r ® (A) of the d-dimensional unitary group are, in general, reducible-even when the representation r of the n-dimensional unitary group is irreducible. We shall treat later the problem of analyzing r ® (X) into its irreducible constituents when r is an irreducible representation of the 2-dimensional unitary group. The irreducible representation of the n-dimensional unitary group whose character is IA), where (X) = (At , , A") is a partition of a positive integer in into not more than n nonzero parts, appears in the analysis of r"% The appearance of this representation tells us that each element of each of the matrices that present, in any basis, the operators of the representation in question is (if it does not vanish) a homogeneous polynomial function, of degree m, of the n2 elements of the corresponding element of

the unitary group. This fact assures us that the representation of the unimodular subgroup induced by this irreducible representation of the unitary group is irreducible. Indeed, if it were reducible, certain homogeneous polynomial functions, f, of degree m, of the elements of the typical

element, X, of the n-dimensional unitary group, would vanish when det X = 1, and this implies that they vanish no matter what the n-dimensional unitary matrix X. Indeed, by virtue of the homogeneity of f, the value off for X is the product of the value off for (exp i8) X, -7 < S< 7r, by exp (-iniS) and we can choose S so that (exp id)X is a unimodular n-dimensional unitary matrix. Since the representation of the n-dimensional unitary group whose character is (A) is irreducible, it follows that the representation of the n-dimensional unimodular unitary group induced

by this representation of the n-dimensional unitary group is also irreducible. 'rhus we have the following result: Each of the functions )A), where (X) _ (At , , A"_t), At > X2 > > 0, is a partition of a nonnegative integer m into not more than n - 1

86


nonzero parts, is a simple character of the n-dimensional unimodular unitary group. The same argument as that used for the unitary group shows that there

are no other continuous irreducible representations of the n-dimensional unimodular unitary group than those whose characters are the functions (A) just referred to. In particular, when n = 2, the only continuous irreducible representations of the 2-dimensional unimodular unitary group are the identity representation, whose character is (0) = 1, the self-representation, whose character is (1) = z1 + z2 = 2 cos a, and the symmetrized mth powers of the self-representation, whose characters are (m),

m=2,3, ; (m)

_

I Z1m+1 1

Z;N+1 I 1

- I z1 1

z2 1

where z1 = exp ia, z2 = exp (-ia), so that (m; = (sin (m + 1)a)/sin a. The value of (m) at the identity point of the parametric space, at which a = 0, is m + 1 so that the dimension of the symmetrized mth power, m = 2, 3, - , of the 2-dimensional unimodular unitary group is m + 1. In particular, the dimension of the symmetrized square is 3 and the charac-

ter of this irreducible representation of the 2-dimensional unimodular unitary group is (sin 3a)/sin a = 1 + 2 cos 2a. In general, (m) = z1m + z1m-1Z2

+ ... + zgm

and, if m is even, this becomes 1 + 2 cos 2a + + 2 cos ma; or, if in is odd, 2 cos a + 2 cos 3a + + 2 cos ma. Whether m is even or odd, we have, p being any positive integer,

(sin (p + 1)a)(m) = sin (p + 1 - m)a

so that,ifp->m,(p)(m) = (p - m} + (p+2-m} +---+(p+m). It follows, on denoting by rp the irreducible representation, of dimension

p + 1, of the 2-dimensional unimodular unitary group, that

rprm = rp_m + rp+2_, + - - - + rp+m ,

p >_ m

Since multiplication of group representations is commutative, this formula furnishes the analysis, into its irreducible constituents, of the product of any two continuous irreducible representations of the 2-dimensional unimodular unitary grup. It is known as the Clebsch-Gordan formula for this group. If we denote by rm the symmetrized mth power of the self-representa-


87

Lion of the n-dimensional unitary group the character of rprm is hphm = (p, m} + hp+ih,,-i

= (p, m} + tp + 1, ne - 1) + hp+2h, _2 and so on. Thus, on denoting by r(a,) the irreducible representation of the n-dimensional unitary group whose character is (Al , A21, we have the relation

r(p..) + r(p+,.arl) + - - - + r(p+m.o)

_ (det X) "`rp-,, + (det x) 'rp+2- +

+ rp+m Since every continuous irreducible representation of the 2-dimensional unitary group is the product of one of the representations r by a power of the determinant representation, this formula furnishes the analysis, into its irreducible constituents, of the product of any two continuous irreducible representations of the 2-dimensional unitary group. It is known as the Clebsch-Gordan formula for this group. -

13

The Spin Representations of the Proper, Time-Sense Preserving

Lorentz Group and of the 3-Dimensional Rotation Group

IfX is any n-dimensional matrix and c is any na X I matrix, the elements of X and c being arbitrary complex numbers, then the (j, k)th element of (X X X)c is vQ,r xy' Q'cCh1 , where xy' denotes the element in the jth row and pth column of X and c" denotes the (p, q) th element of c, p = 1, , n; q = 1, - - , n. On denoting by C the n-dimensional matrix whose element in the pth row and qth column is cPQ, it follows that the (j, k)t.h element of c' _ (X X X)c is the element in the jrh row and lath column -

of XCX,* so that the correspondence c -+ c' = (X X X)c may be conveniently presented in the form c - C' = XCX*. It follows that there

exists a basis in which the matrices that present the linear vector operators, which are presented by the matrices X X X, are all real. Indeed, the relation C' = XCX* implies the relation C'* = XC*X* and, hence, the relation (C' - C'*) = X(C - C*)X*. The matrices (C - C*) and (C' - C'*) are elements of the n2-dimensional linear vector space whose elements are n-dimensional anti-Hermitian matrices and so, in presenting the correspondence c -+ c' = (X X X)c, we may restrict ourselves to those n' X I matrices c which correspond to n-dimensional anti-lIermitian matrices C. Now, the linear vector space whose 88


89

elements are n-dimensional anti-Hermitian matrices has a basis for which the coefficients of the linear combinations involved are all real, and in this basis the linear operators that are presented-in the basis whose elements are n-dimensional matrices the elements of which are all zero save one-by

the matrices X X X, will be presented by real matrices. EXAMPLE: n = 2

The general 2-dimensional anti-Hermitian matrix is of the form

i(d" + dP) - d12 + ids'

d12+01-i(d"-d22)I where

d"

d'2

d = d2'

Id"I is a real 4 X 1 matrix. The equations,?' = i(d" + d2s), c12 = -d'2 + ids', etc may be written in the form c = Pd, where P is the 4-dimensional matrix

00 i -1 i 0

i

P=

0 0

-i

i

1

0

00 i

and the transformation c -+ c' = (X X X)c appears, in our new basis, which is determined by the four 2-dimensional anti-Hermitian matrices

M,=(0 -OSI,

A12=I1

Ma

OI,

l0

iI,

M4

.1i 0I

as d -+ d' = P-'c' = P-1 (X X X)c = P-'(X X X)Pd, the matrices P-'(X X X)P being all real. It will be important for us, in a moment, to

know that the element in the 4th row and 4th column of P(X X $)P is not only real but positive, no matter what is the nonzero 2-dimensional complex matrix X. To see this we observe that 2-iP is unitary, so that -i

P-'

P'`

0

i

0 -1 1 0 -i -i

0 0

-i

0

0

0

-i

Thus the first and last elements of the fourth row-matrix of P-'(X X 1)

are - (x,'x,' + xz'z,2) and 2

(xs'x21 + x2W), respectively, so that the 2


00

element in the fourth row and fourth column of P- (X X ±)P is (x11xl' + xi2x12 + x2'x2' + x2 x_2 ),

which is positive.

The matrices P-'(X X X)P furnish, by means of the correspondence X --+ P-'(X X X)P, a real presentation of an n2-dimensional representa-

tion of the n-dimensional linear group, this representation being the product of the self- and conjugate-representations of the group. It follows

from the relation C' = XCX* that the determinant of C' is the product of the determinant of C by the squared modulus of the determinant of X. If, then, we restrict our attention to the unimodular subgroup of the ndimensional linear group, the determinant of C is invariant under all the linear transformations c -, c' = (X X X) c. When n = 2, the determinant of

2(d" + dd) - d12 + id2' G' = I d12 + id2i - i(d11 - d22) is

(d")2

+

(d12)2

+

(d21)2 - (d22)2

Writing d" = d', d12 = d2, d2' = d3, d = d' we have, then, the following result:

The real transformations d --' d' = P-' (X X X) Pd, where X is any unimodular 2-dimensional matrix, leave invariant the quadratic form (d1)2 + (d2) 2 + ([LL,3)2

(,1)2 [[ll

in Ilic elements of the 4 X 1 matrix d. We term any homogeneous linear transformation, d d' = Ld, which leaves the quadratic form (d')2 + (d2)2 + (d3)2 - (d')"- invariant, a

homogeneous Lorentz transformation, and we term the matrix, L, which presents the homogeneous Lorentz transformation, a Lorentz matrix. On denoting by F the 4-dimensional diagonal matrix all of whose diagonal elements are 1 save the last, which is -1, a 4-dimensional matrix, L, is

a Lorentz matrix if, and only if, L*FL = F, just as a 4-dimensional matrix, 0, is an orthogonal matrix if, and only if, 0*0 = E4 or, equivalently, O*E,O = E4. Thus F plays for Lorentz matrices the role played by E4 for orthogonal matrices. The same argument as that used for orthogonal matrices shows that the collection of Lorentz matrices constitutes a group, the Lorentz group, and that, if L is a real Lorentz matrix, det L is either 1 or -1. The collection of unimodular real Lorentz matrices constitutes a subgroup of the Lorentz group; we term a unimodular real Lorentz matrix a proper Lorentz matrix, so that this subgroup is the group of all proper Lorentz matrices. If L is any real Lorentz matrix, proper or not, (141)2 + (la2)2 + (143)1

- (l; )2 = -1 so that 114' 1 > 1, the equality

holding only when 14' = 0, 142 = 0, 143 = 0. The relation L*FL = F im-


91

plies the relation 11* = FL-'F and so, since the collection of Lorentz matrices constitutes a group, to which belongs the matrix F, L* is a Lorentz matrix when L is. If, then, 114' 1 = 1 not only are 14', 142, l43 all zero but 11', J_' and 1,' are also all zero and L is of the form 03

0

0

±1

where 03 is a 3-dimensional orthogonal matrix. Since (144)2 = 1 + (141)2 + (142)2 + (143)2

it follows that, if l4' is positive, it is greater than 1(14')2 + (142)2 + (14x)21;

and this implies, on dealing with L*, that it is also greater than 1 (114) 2 + (12') +

(12,4) 21'.

If, then, L and L' are any two real Lorentz matrices, proper or not, for each of which the element in the 4th row and 1th column is positive, then the element in the 4th row and 4th column of their product, L'L, is also

positive; for this element is 14''14' + 12'l42 + 1,"14' + 1411144 and 111"14' + 12'142 + 124143

< ((11P1)2

+ (1:1)2 + (1311)21' 1 (14') 2

+ (142) 2 +

(l43) 21' < 14 4144.

Thus the collection of real Lorent.z matrices, L, for which 144 is positive and,

hence, > 1, constitute a subgroup of the Lorentz group. We term any real Lorentz matrix, L, for which 144 is positive, it time-sense preserving Lorentz matrix, and we term the collection of all time-sense preserving unimodular Lorentz matrices the proper, time-sense preserving, Lorentz group. We

have seen that the matrix L = 13-'(X X X) P, where X is any 2-dimensional unimodular matrix, is it Lorentz matrix and, moreover, since its determinant is the square of det X det ?, it is a proper Lorentz matrix. We have seen, also, that, it is it time-sense preserving Lorentz matrix and so we have the following result: The collection of matrices P-' (X X X not being unambiguously determinate, we say that it is a 2-valued, or spin, representation of this Lorentz group; similarly, we say that the 2-dimensional unimodular unitary group is a 2valued, or spin, representation of the 3-dimensional rotation group. What we really mean by these statements is that the 2-dimensional unimodular group is a representation of the doubly covered, proper, time :sense preserving Lorentz group and that the 2-dimensional unimodular unitary group is a representation of the doubly covered 3-dimensional rotation group-just as, in the theory of functions of a complex variable, zi is not, properly speaking, a function of the complex variable z, but rasher of the two-sheeted Riemaun surface which is the doubly covered complex plane. Any representation r of dimension d of the 2-dimensional unimodular group in which Ed corresponds to -E2 is a representation, in the strict, sense, of the proper, time-sense preserving Lorentz group while, if Ed does not correspond to -E2 , it is a spin representation of this Lorentz group; similarly, any representation r of dimension d of the 2-dimensional unimodular unitary group in which Ed corresponds to -E2 is a representation, in the strict sense, of the 3-dimensional rotation group while, if Ed does not correspond to -E2 , it is a spin representation of the threvdimensional rotation group. On the other hand, all representations of the proper, time-sense preserving Lorentz group, or of the 3-dimensional rotation group, are representations of the 2-dimensional unimodular group, or of the 2-dimensional unimodular unitary group. respectively, in which

the identity matrix of the representation corresponds to -E2. Thus, every continuous irreducible representation, proper or spin, of the 3-dimen-


05

sional rotation group is one of the symmetrized powers, whose characters are the functions) tit) = hm , of the self-representation of the 2-dimensional unimodular unitary group. When in is even, the matrix of the representation, of dimension in + 1, which corresponds to -E2 , is Em+i ; when tit is odd, the matrix corresponding to -E2 is -Em+, ; indeed, each element of each of the matrices which present the linear transformations of the representation whose character is 1m) is a homogeneous function, of

degree in, of the elements of the 2-dimensional unimodular matrix to which it corresponds, and this implies that -Es - (-1) mEm+c , since f%- - Em+t . Thus the only continuous irreducible proper representations of the 3-dimensional rotation group are the representations of the 2-dimensional uniinodular unitary group, whose characters are the functions (0), (2), (4), . These are of dimensions, 1, 3, 5, - - and we denote by r;+, j = 1, 2, , the representation, of dimension 2j+1, whose character is )2j), the identity representation being denoted by ro. Similarly, the only continuous irreducible spin representations of the 3-dimensional rotation

group are the representations of the 2-dimensional unimodular unitary group, . These are of dimensions whose characters are the functions {1), )3), )5), 2, 4, G, and we denote by r; , j = 1, 2, , the spin representation of dimension 2j, whose character is 12j - 11. From the Clebseh-Gordan formula for the 2-dimensional unimodular unitary group we obtain the following analyses: -

r,+rk+ = r k + 1' k+i + ... + r k

k

r,rk =r

k+rfk4.1+...+r,+k_I, j> -k r; rk+ = r;_k + rJ-k+c + + I +k , j > k

r, rk+ = rk-,.,.1 + r--j+- + ... + rk+, , k > j 'i'hesc are the Clebseh-Gordan formulas for representations, both proper and spin, of the 3-dimensional rotation group.

The fact that the proper, time-sense preserving Lorentz group is a representation of the 2-dimensional unimodular group assures us that all continuous representations, proper or spin, of the proper, time :sense preserving I.orentz group are to be found among the continuous representations of the 2-dimensional unimodular group. The theory of the representations of the 2-dimensional unimodular group is more complicated

than that of its unitary subgroup, the reason being that the parametric space of the 2-dimensional unimodular group is not bounded and closed. 't'hus we are barred from the process of averaging over the group which proved so useful in the theory of the representations of the n-dimensional unitary group. Hence, for example, we have no assurance that, all con-

96


tinuous representations of the 2-dimensional unimodular group are unitary; on the contrary, it is easy to see that the only irreducible unitary representation of the 2-dimensional unimodular group is the identity representation. Indeed, the argument used in the case of the n-dimensional unitary group shows that, if r is any unitary representation, of dimension d, of the 2-dimensional unimodular group, the characteristic numbers of the matrices which present those operators of r which correspond to the 0 ,of the group are integral powers, positive negadiagonal elements,IO tive or zero, of z and, when I z 1 0 1, the only integral power of z whose modulus is unity is the zero power. Thus the matrices which present the

operators of r which correspond to those diagonal elements Iz

0 iI, of

the 2-dimensional unimodular group for which I z I qd 1 are each the d-

dimensional identity matrix, Ed. Since r is continuous, it follows that the matrices which present the operators of r which correspond to the elements of the diagonal subgroup of the 2-dimensional unimodular group are each the d-dimensional identity matrix Ed. The same argument shows

that the matrices which present the operators of r, which correspond to

ll, s variable, or to the elements of the

the elements of the subgroup, IO subgroup,

11

0i, s variable, of the 2-dimensional unimodular group, are

each the d-dimensional identity matrix, Ed. Every 2-dimensional unimodular matrix, Ib

di,

for which a 0 0, may be written in the form 1 11 I

0

a-'

01 ll

8 where 8 = ca', s' = ba 1, and so the matrices which present the operators of r which correspond to those elements of the 2-dimensional unimodular group for which a 0 0 are each Ed. Since r is continuous, it follows that all the matrices which present the operators of r are each Ed and, when r is irreducible, this implies that d = 1 and that r is the identity representation. Thus the proper, time-sense preserving, Lorentz group does not

possess any continuous unitary representation other than the identity representation.

Nonunitary representations of a group do not, in general, possess the convenient feature of unitary representations that the existence of an invariant subspace of their carrier space implies the existence of a second

invariant subspace of this carrier space, and so the concept of addition of representations is not applicable. We say that a nonunitary


97

representation is irreducible when its carrier space does not possess an invariant subspace. For example, the self-representation of the commutative subgroup of the 2-dimensional unimodular group which consists of the matrices IO

111 s variable, is reducible, since

1I0

1I

el = c1, every 8,

but this representation cannot be written as the sum of two 1-dimensional representations. The various symmetrized inch powers of the 2-dimensional unimodular group furnish irreducible representations of the proper, timesense preserving Lorentz group; when m is even, the dimension, in + 1, of the representation is odd and the representation is a proper representation of the Lorentz group in question. On the other hand, when in is odd, the dimension, m + 1, of the representation is even and the representation is a spin representation of the Lorentz group. The irreducibility of these various representations of the proper, time-sense preserving Lorentz group is an immediate consequence of the irreducibility of the corresponding representations of the unitary subgroup of the 2-dimensional unimodular group.

14

The Simple Characters of the n-Dimensional Rotation Group

We consider first the case when n = 2k is even. The 2k characteristic mmibers of the typical element of the 2k-dimensional rotation group are of the form zk+l = exp - ia, , , z, = exp ial , , zk = exp iak ,

zu = exp - iak where a, , , ak are the class parameters, and it follows that the polynomial, f(t), of degree n in an indeterminate t:

oo - o,t + ... + (- 1)iCrntn

f(t) = (1- zit) ... (I is such that o2k-j = of

0, 1,

= 00 - o,t + ... + o2kt2 , k - 1. Indeed, f(t) = Jj;_,(1 - z,t)

(1 - -*It) = Jj;a,(2i - 1)(zj - 1), since zjzj = 1, and so Of(] /t) = f(t), which implies that o2k = ou , o2k-, = o, , . when t is any one of the 2k numbers z, ,

-

, ok+, = 0k_, . Since f (t) = 0 , zk, z, , , 2k we find, on

, k, that oo'2j" + o,'ij"-' + ... + o", + o"+,zj + ... +

multiplying f(zj) by ij", p = 1, where o,' =

r = 0, 1,

QO xj" + ul'zr' +

=0

, 2k. Similarly,

+ Q" + 0"+12j +

and, since QYk_V = o,', q = 0, 1,

o2&Z2k-v

+ 620

"=0

, k - 1, 2k - q being even or odd 98


99

according as q is even or odd, respectively, we obtain, on subtraction, the relation Oo (z,2k-P _ zj2k-P)

+ _L ... + QP+1(zj - Zj) + P-1(zj - z,) + .. . + 00 (zjP - zjP) = 0 On denoting z' - z by sq(a), where z = exp ia, so that 8q(a) = 2i sinqa this relation appears as uo 82c-P(aj) + ... + D'P+tsi(aj) = QP-isl(aj) + ... + Oo's,(aj),

j Denoting, for a moment, by s(a) the 2k X 1 matrix whose qth element is s2k_9(a), q = 1, - , 2k, this relation states that the pth and (2k - p)th elements of the 2k X 1 matrix T2ks(a) are the same, their common value being the (k- p + 1)st element of the k X 1 matrix sk(a) Tk

:

s1(«) is the zn-dimensional triangular matrix whose rth row-matrix is (at_,, OlZ_, , , Olin-,), it being always understood that any Q with a negative subscript is zero.We have already seen, when diszk eu& ing the unitary group, that, if z is any one of the 2k numbers z1 , z1 , , ik, and if l is any nonnegative integer, then

where T., in = 1, 2,

,

_2k- I

Zt

= (hl)f'rk

e-21 11

1

where (hz) is the 1 X 2k matrix (hz-2k+1, , hz). Substituting, in turn, zj and Zj for z and subtracting, we obtain the relation

st(aj) = (ha)T2ks(aj) Since the last element of the 2k X I matrix T2ks(aj) is zero, being the sa its the lust element of the 2k X I matrix s(aj), and since the pth and (2k -

p)th elements of Tus(aj) are the same as the (k - p + 1)th element of sk(aj)

it follows that

Tk

81(aj) sk(aj)

al(ai) = (hz-k, hz-k-1 + hz-k-1-1, ... , ha-tit+1 + ht-1)Tk 81 (aj)


100

If the nonnegative integer l is positive we may replace 1, in this relation, in turn by d + 1 and 1 - 1 and we obtain, on subtracting the two resulting equations, the equation

81(ai)ct(ai) = (hi-k+i, hi-k + hi-k+2, .. , hi-2k+2 + hl') Tk

8k(a,) :

st(aff)

=2cosla,11,

wherehm' =h,,

. This equation remains valid when l = 0, since it reduces, when

2,

1= 0, to 18k(ai)

81(ai) = (0, 0, ...,0, 1) Tk I

= (0, 0, ..,0, 1)

81(ai)

!l(at) = 81(ai) 81(aj)

co(a) being 1. Now let (X) = (Al , - , Ak), where x, > A2 > > Ak> 0, be any partition of any nonnegative integer m into not more than k nonzero

parts and set l1 = Al + k - 1, is = A2 + k - 2,

-, lk = Ak , so that -> 4 ? 0. On assigning, in turn, to 1, in the equation which l1 > 12 > furnishes s, (ai) ct(a,) , the values 11, £2 , - , 4, we see that the product of the k-dimensional diagonal matrix whose diagonal elements arc sl(at), , s1(ak) by the k-dimensional matrix whose pth row-matrix is (ct,(at), ,

ct,(ak)) is the product of three k-dimensional matrices which are,

reading from left to right, (1) The k-dimensional matrix whose pth row-matrix is (hi,-k+t , ht,-k + , hi,-2k+2 + hi,) (2) Tk

(3) The k-dimensional matrix whose jth column-matrix is 8k(at)

st(aff)

Of the five matrices involved, only (1) and the matrix whose pth row-matrix is (ct,(at), - , et,(ak)) depend on the numbers (l1, - - , 4) or, equiva-

lently, on the numbers (A1, - , Ak) and we denote the determinants of these two matrices by [A] and C(l), respectively. Thus [A) is the product of C(l) by a factor which does not vary with (A) ; to find out what this factor is we set (A) = (0, 0, - , 0) so that (1) = (k - 1, k - 2, - - , 1, 0). When we do this the matrix (1) reduces to a triangular matrix whose diagonal elements are all ho' = 1, so that [0[ = 1. Hence the desired factor is the , 1, 0) : reciprocal of C(k - 1, C(l) ' 1, 0) [Al

- C(k - I.

The symmetrized mth power of the self-representation of the 2k-di-


101

mensional unitary group is an irreducible representation, r, whose character is hm , of the 2k-dimensional unitary group. P induces a representation,

in general reducible, of the 2k-dimensional rotation group, and so h,,,' = hm - h.-2, in = 0, 1, 2, , is a generalized character of the 2k-dimen-

sional rotation group, and this implies that [A), the determinant of a k-dimensional matrix whose elements are linear combinations, with integral coefficients, of the functions hm', is a generalized character of the 2k-di-

mensional rotation group; we proceed to find the average of its squared modulus, i.e., of its square, over the group. Since the normalized class factor of the element of volume of the 2k-dimensional rotation group is { 1/k ! (2a) k ) IC(k - 1, , 1, 0))2d(a) the desired average is the product of ft C(I)) 2 , d(a) by 1/k!(2,r)k. The argument which showed that f{C(k - 1, 1, 0))2d(a) = k!(2r)k shows that if lk = 0, f IC(t))2d(a) = k!(2a)k while, if lk > 0, f{C(l))2 d(a) = 2(k1)(2ir)k, the reason for the difference in the two results being that ci(a) = 2 cos la, if 1 > 0, while co(a) = 1, so that the integral of {co(a)12 over the interval 0 < a < ,r is only one-half the

integral of [c,(a))2, l > 0, over the same interval. Since lk = Ak we have, then, the following result: The average of [A)2 over the 2k-dimensional rotation group is 1 if Ak = 0, and 2 if AA; > 0.

Our next step is the evaluation of [A) at the identity element of the group,

where all the class parameters a, , . , ak are zero. Treating the a's as infinitesimals, the lowest order terms in C(l) are furnished by the determinant of the k-dimensional matrix which is obtained from the k-dimensional matrix whose determinant is C(l) by replacing each element, c,(a), of the latter matrix by the sum, ,2k 2a2k-2 ,a4 12x2 -1- (-1)

2 (1 -

2!

+

(2k

f!

- 2) !}

if l > 0, or 1, if l = 0, of the first k terms of its development as a power series in a2. This modified matrix may be written as the product of two k-dimensional matrices which, reading from left to right, are (1) The k-dimensional matrix whose pth row-matrix is (1, lp , P= 1 , , k, if lk > 0; if l k = 0 the kth row-matrix is 4 ( 1 , 0, 1, 1k2, ... , lrk-2) . (2) The k-dimensional matrix whose jth column-matrix is 1

,

2k-2)

1P

,

0) _

102


multiplied by 2. The determinant of the matrix (1) is (- 1) k(k-" 12 times the difference product (11 - 122) . . (Ik_1 - lk) of the k squared integers ,1k, if lx > 0, and one-half of this if lx = 0, and so the limit, as Ik, (al, ... , at.)

C(li, ... , 4)

(0, 0, ... , 0), of [A] =

1, 0)

is, if lx = 0, the quotient of the difference product A(12) = (112 - 122) (11_1 - 1k) of the k squared integers 112,

2-(x-')(2k - 2)1(2k - 4)1

..

-

,

1x2, by the difference product,

2!, of the k squared integers (k - 1)2,

(k - 2)2, - , 1, 0 and twice this if is > 0. Since [AJ is a continuous function of the k-class parameters a1, - , ax , it follows that the value of [AJ at the identity element of the 2k-dimensional rotation group is 2k-',&(12)

(2k- 2)1(2k - Q! .. 2 ! 2ko(t2)

d

and (2k - 2)1(2k - 4) ! .. 2!

lk

if

=

0

lx > 0

Since these values are positive, we have the following partial result: If Ax = 0, [A] is a simple character, i.e., the character of an irreducible representation, of the 2k-dimensional rotation group, the dimension of this

irreducible representation being 2k-'A(12)/(2k - 2)1(2k - Q! ... 2! The various irreducible representations of the 2k-dimensional rotation group which we obtain in this way are all different, since their characters are different. When k = 1, the only irreducible representation of this type is the identity representation, for which m = 0, (A) = 0 and [A] = 1; when k > 1 we have, among the irreducible representations of this type, one

whose character is h,,,' = h.- h.-:, for each nonnegative integer in. When rn = 1, we obtain the self-representation whose character is hl' = Its .

When m = 2, we have the relation 121 = h2 - ho = (2) - (01 or, equivalently, the relation 12) _ [21 + [0]. Thus the symmetrized square of the self-representation of the 2k-dimensional rotation group, where lc > 1, is reducible, one of its two irreducible constituents being the identity representation. Similarly, [3] = 13) - (1), so that (3) = [3] + [l]; [4] = (4) 12), so that 14) = [41 + 121 + 10] and, generally, (in) = I11t] + (m - 2) _ 1m1 -l- [m - 2] -l- (m - 4) = [m] -l- [»a - 21+ [m - 4]rr ... where the summation on the right ends with [1) or [01, according as m is odd or even, respectively. This formula furnishes the analysis of the symmetrized With power of the self-representation of the 2k-dumensional rotation

group, where k > 1, into its irreducible constituents. When Ax > 0, [A] is not a simple character of the 2k--dimensional rotation group since the average of its squared modulus, i.e., of its square, over the


103

group is 2, rather than 1. Since the value of [NJ at the identity element of the group, namely 20(12)/(2k - 2) ! 42! , is positive [XI is either the sum of two simple characters or the sum of a simple character and the negative of a simple character. In either event it contains a simple character, which we denote, for the moment, by X. Since X is a class function, it is a symmetric function of the k-class parameters a,, , ak which is, more-

over, unaffected by a change of sign of an even number of these class parameters. Hence C(k - 1, , 1, 0)X is an alternating function of the k-class parameters which is unaffected by a change of sign of any even number of these class parameters. On the other hand,

C(k - 1, ..., 1, 0)[X] = C(1) = F, f exp i(f1P,a, f ... f 1Pkak) (P)

is all even alternating function of the k-class parameters. There are A! terms in the expansion of C(1) and half of these have an even number of minus signs in the argument of the exponential, the other half having an odd number of negative signs in the argument of the exponential. If we change, in the expansion of C(1), the signs of this latter group of 2k-'k! terms we obtain the expansion of S(1), the determinant of the k-dimensional

matrix whose jth row-matrix is (s:, (a,), - , s,1(ak)). Thus the sum of the 2k-'k! terms of C(1) for which there are an even number of negative signs in the argument of the exponential is if C(1) + S(1) J and the sum of the remaining 2k-'k! terms of C(1), each of which has an odd number of negative signs in the argument of the exponential, is z[C(1) - S(l)J. If C(k - 1, , 1, 0) X contains a single term of the first set it contains them all, since it is an alternating function of the k-class parameters which is unaffected by a change of sign of any even number of them. Similarly, if C(k - 1, - , 1, 0) X contains a single term of the second set it contains them all and this implies that if C(k - 1, , 1, 0) X contains a term of either of the two sets it does not contain any term of the other. Indeed, if it did, it would coincide with C(1) = C(k - 1, , 1, 0)[XJ so that X would coincide with [XJ, which cannot be, since X is, by hypothesis, a simple character of the 2k-dimensional rotation group, while [N] is not a simple character of this group. Thus C(k - 1, .. , 1, 0) X iseither 11C(1) + S(1) J or 1[C(1) - S(1)]. The value of S(1)/C(k - 1, , 1, 0) at the identity element of the group is zero, the terms of lowest order in the infinitesimals a, , , ak in this quotient being a constant times the product of these , 1, 0), which we infinitesimals. Hence both [C(1) + S(l)J/C(k - 1, shall denote by @J+, and a1C(1) - S(l))/C(k - 1, , 1, 0), which we shall denote by [AJ_, have the same positive value, namely, 2k-'A(l2) / (2k - 2) !(2k - 4) ! 2! at the identity element of the group. It follows, since [XI = [X]+ + @J- , and since one or other of the two quantities [XJ+

104


and [A]_ is the simple character X, that both [A]+ and [A]_ are simple charac-

ters of the 2k-dimensional rotation group. Thus when Ak > 0, [A] is a compound character of the 2k-dimensional rotation group, being the sum of two simple characters [A]+

#(C(l) + S(1) J

C(k - 1,

, 1,0)'

[A]- _

I[C(l) - S(l) J C(k - 1,

, 1, 0)

the common dimension of the two irreducible representations of the group of which these are the characters being 2"-'A(ls)/(2k- 2) ! . . . 2! Note. This result generalizes do Moivre's theorem, exp(f im0) = cosm0

f i sin m0, to which it reduces for the 2-dimensional rotation group, , 1, 0) becomes 1, (A) becomes for which k = 1. When k = 1, C(k - 1, m, C(l) becomes 2 cos 10 = 2 cos m0 and S(1) becomes 2i sin m0. Thus the

symmetrized mth power of the self-representation of the 2-dimensional rotation group, m > 0, whose character is [m] = 2 cos m0 is reducible, being the sum of two 1-dimensional representations whose characters are [m]+ _

exp into and [m)_ = exp (- imO). The same argument as that used in the case of the n-dimensional unitary group shows that the character, X, of any continuous representation of the 2k-dimensional rotation group is a linear combination, with positive integral coefficients, of terms z,°' zk'k where the s, , , sk are integers, positive, negative, or zero. Thus C(k - 1, - , 1, 0)X is, by virtue of the fact that it is an alternating function of the k-class parameters a,, - , ak , which is unaffected by an interchange of sign of any even number of them, a linear combination, with positive integral coefficients, of the functions C(l) and

J[C(l) f S(t)), the latter functions not appearing if X is an even function of all the class parameters. Hence There are no continuous irreducible representations of the 2k-dimensional

rotation group other than those whose characters are the functions [AJ, [AJ+ and [A]_ , the two latter appearing when, and only when, Ak > 0, , Ak) being any partition of any nonnegative integer m into (A) = (X1 , not more than k nonzero parts.

We now turn to the case where n = 2k + 1 is odd. The characteristic numbers of any (2k + 1)-dimensional rotation matrix are of the form , zk , Z, , . , ?k) and we denote by ho*, h,*, (1, z, , symmetric functions of the last 2k of these. Thus

the complete

11(01 -1 =0 - t)-1((l -919)(1 -fit) -0 - zkt)(1 - 401

1-t Since If(t) J-' = ho + hit +

,

where ho , h, ,

are the complete sym-


105

metric functions of all 2k + 1 of the characteristic numbers, we have (1t2) (T(1) -, = he' + hc't -l- h2't2 + .. , where hm' = hm - hm-2 , m = 0, 1, 2, . , and so (1 + t) I he* + h,*t + I = ho' + h,'t + . Hence

ho' =ho*= 1,

hj'=hj*+hj 1, j- 1,2,.

We have seen, when discussing the 2k-dimensional rotation group, that 8k (aj )

sl(a;) _ (hi k, hi k-c -I- hi k+i, ...

,

hl**-2*+i -i- hi 0Tk

81(aj)

and it follows that 2 Cos 2 81.1(aj) = sl(aj) + sl+t(aj)

sk(a!)

_ (h;-k+c, hl-k + h,-k+2, . . , hi-2k+2 + hl')T1

s,(ai) Thus S(1 + 4), the determinant of the k-dimensional matrix whose pth s1,+,(ak)), is the product of [A], the row-matrix is (sl,+;(a,), determinant of the k-dimensional matrix whose pth row-matrix is (hi,-k+l

,

h,-k + hl,-k+2, h',-2k+2 + hi,), by a factor which is independent of (l) or, equivalently, of (A). Setting (A) _ (0, 0, 0, - - 0) we have [AI = 1, so that the multiplying factor is S(k - 4, Thus

Since the normalized class factor of the element of volume of the (2k + 1)-

1, 4) I2/k!(2ir)k]d(a) it follows, by the argument used in the cases of the n-dimensional unitary,

dimensional rotation group is p S(k - ;,

-

,

and the 2k-dimensional rotation, groups, that the average of IIX)12 over the

(2k + 1)-dimensional rotation group is 1. To show that [A] is a simple character of the (2k + I)-dimensional rotation group we have only to show that the value of (A] at the identity element of the group, where the k-class parameters are all zero, is positive. Denoting t + # by l', the lowest order terms in the infinitesimals at , , ak which occur in the expansion of the determinant S(1') are obtained by replacing each element, 8,,(a), of the k-dimensional matrix of which S(1') is the determinant by the sum 2il'a (1 - (1j2a2/3 !) + ... + (-1) k-' (l'2k-2a2k2/ (2k - 1) !) ) of the first k terms of its development as a power series in a. Thus, by the argument used in the case of the 2k-dimensional rotation group, the value of [A] at the identity element of the group is the quotient of l; lk times the difference product of the k numbers llj2, , lk'2 by (k - #) 1 times the difference

100

Lectures on Applied illathematies

product of the k numbers (k - 4-)2, p+j)2- ( k-q- } 2)2[,p

,

(%)2. Since the product IIq[k

-

Im + j - 11,

{p)(m+j-1) -{p)(m+j-3)

={m+j+p-1)+{p-m-j+1) while, if p<m+j-3,

{p}(m+j- 11 -(p)(m+j-3) = (m+j+p- 11 - (m+j - p- 31 and, finally, if p = m + j - 2,

(p){m+j-1)-(p)(ns+j-3) =(m+j+p+1) For example, since {2) 0 (2) = {4) + 1011,

{3) 0 (3) =(3)+{3)+(9)+{5)-{3) =(9)+{5)+(3) Similarly, since (2) 0 (3) = (3) X (2) = {6) + (2),

(3) 0 (4) = {4) + (6) + (2) + (12} + (0) + (8) - (2)

=(i2)+(8)+(6)+(4)+(0) and, since (3} 0 (3) = {9) + (5) + (3) and

(2) 0 (4) = (4) 0 (2) _ {8) + (4) + (0), (4) 0 (4) = 2(8) + 2(4) + 2(0) + (16) + (2) + (12) - (0) + {10) - (2) = (16) + (12) + (10) + 2(8) + 2(4) + (0) and so on.


113

In order to analyze r(,) ® (X), where (A) = (m - 1, 1) is a 2-part partition of in whose second part is 1, into its irreducible constituents, we observe

that the relation (rn - 1} (1} _ (m} + (m - 1, 11, applied to the (p + 1)dimensional unitary group, p + 1 being the dimension of r(,), tells us that the product of (p} ® (m - 1) by (p) is the sum of (p} 0 (m) and

(p}®(m-1, 1) Thus

(p)®(m-1,1)=((p}®(m-1))(P}-IM ®(m) For example,

(2} 0 (21) = (141 + 101) (2) - (6) - (2} = (4} + (2} We have seen, when proving Hermite's Law of Reciprocity, that

{P1 0 (m) = ((P - 11 0 (m))z2m + ({Pl ® (m - 1))z,' and we obtain from this relation, on multiplying through by z, , the relation

({P) 0 (m))z1 = ((P - 1) ®

(m))z2m-`

+ ({P) ® (in - 1))z,"+'

Interchanging z, and z2 in this relation, subtracting and dividing by z, - z2, we see that

{P) ®(m) so that

-((p - If ®(m)) {m - 2) + ({P) ®(m - 1)) (pl

(p) 0 (m - 1, 1) = ((p - It 0 (m)) (m - 2)

For example

(3) 0 (4, 1) _ (1101 + {6) + 121) 131

= (13} + (11) + 2}9} + 2(7} + 2(5} + 2(3) + (1} The analysis of the ant isymmetrized mth power of r(,) into its irreducible constituents may be obtained from the useful relation

(P) ®(lm) = fp+1 -nil ®(m), p= 1,2, --It is clear that this relation is valid for every positive integer to if p = 1, both sides vanishing if tit > 2 and both sides reducing to (0) if m = 2 and to (1 } if m = 1. We prove the validity of the relation for values of p > 1 by induction; i.e., q being any positive integer, we show that the validity of the relation, for every positive integer m, when p < q, implies its validity

when p = q. Since (p) 0 (1 t) is the elementary symmetric function of degree j of the p + 1 numbers z,', z,p-'z2,

, z' the polynomial function

of the indeterminate t, f(t) = (I - zipt)

(1 - 22pt), is

1 -(Plt+({P) ®(12))12-...


114

and it follows, since f(t) is the product of (1 - z1r"'t') where t' = z21, by 1 - z1't, that

(1 - z2'-'t'),

{p} ® (11 = ({p - 11 0 (tm))z2m

+ ({p - 11 0

(lm-')):1P-m+1,

In < p +

1

and from this relation we obtain, on multiplication by z1 , the relation

((p{ 0 (lm))z1 = ({p - 1) 0 (1m))z2m + ({p - 1) 0

(1""-'))Z1'_"m_2

On interchanging z1 and z2 in this relation, and subtracting the relation thus

obtained from the relation just written and then dividing the result of the subtraction by z1 - z2, we see that ®(1'"-1))

{p) 0 (1m) = ({p - 11

- ((p -

{p - m + 1) 11 0 (1m))

(,, - 2), m < p + 1

Setting p = q and using the hypothesis of our induction proof we have

(lm) = ({q - in + 11

(q)

(in - 1)) {q -m -l- 1)

- ((q - ml 0 (m)){m-21, m 0, the last column-matrix of T;,, +k is not erased and

the last column-matrix of the A,-dimensional matrix which remains after the erasures is

Qk

Counting from its last column-matrix the first column-matrix of T.\, +k which is erased is the (Ak + 1)th, and so the subscripts of the last Ak diagonal elements of the A,-dimensional matrix whose determinant is ±( A) are all equal

to k. The second column-matrix erased, counting from the last columnmatrix, is the (Ak_, + 2) th so that the next diagonal subscript, counting upwards to the left, of the A,-dimensional matrix that remains after the erasures is less than k by the number of X's that are equal to Ak . Reasoning in this way, we see that the diagonal subscripts of the A,-dimensional matrix whose determinant is +{A}, reading upwards to the left, are the elements

of (,\*), the partition of m which is associated with (A). Reflecting this A,-dimensional matrix in its secondary diagonal, we do not change its determinant; a change of sign of all column matrices which have a o' with an odd subscript at the bottom followed by a change of sign of all rowmatrices that have a o' with an even subscript at the end removes the primes

from the o"s, the only cost being a possible change of sign of the determinant of the matrix. After the reflection in the secondary diagonal the subscripts of the diagonal elements, reading downwards to the right, are the elements of (A*), so that the determinant of the A,-dimensional matrix is {A*), . Thus {A} = f {A*}, , where the ambiguous sign is independent of n and of z, , , zp . When z, , - , z are all set = 1, { A) is positive, its value being the dimension of the irreducible representation of the n-dimensional

unitary group whose character is {A), and so we shall know that the am-


117

higuous sign is + if we can show that {A*}, is positive at the identity point, e, of the parametric space. When k = 1, (A*J, = (1"'J, = (mJ is positive at e; when k 2, we take n=2so that, at e,03= I.. Since A,* < Al* = 2, we see that, at e, (A*J, is positive, having the same value as { t'J, for some nonnegative integer j. Hence, when k = 2, {A} = (A*J, , no matter what is the value of n. When

and {A*J, = {Ay*, A,*,

k _ :1 we proceed in the same way, taking n = 3, so that, at e, 03 = 1, and we find that { AJ = { A*J, no matter what the value o4 = 0, ob = 0, of n. In this way we prove that {A} = (A*J, for every (A). The functions {p} 0 (1'), j = 1, 2, , p + 1, are the elementary sym-

metric functions, E1, , Ea+1, of the p + I characteristic numbers, , zzr, of a typical matrix of the irreducible representation,

z,', ZIP-'Z2 ,

1'(,) , of the 2-dimensional unimodular unitary group, and it follows, since z,'-'z2' is the reciprocal of z1'z2"', j = 0, 1, , p, that E1 = Ep+,-j . The relation (A*} = {AI, tells us that {pJ 0 (A*) is the determinant of the k-dimensional matrix whose jth row-matrix is -

-

(E,1-k+l Er1-k+2 , ' - . ,E11) (A) being a k-part partition of any positive integer in and l; being X j + k - 1, j = 1, - - , k. On denoting by Ai-f}1 the complement, p + 1 - Aj, of X j in p + 1 and using the relation Ej = E +1- j , we see that the diagonal elements of the k-dimensional matrix obtained by reflecting the kP

dimensional matrix whose determinant is {pJ 0 (A*) in its secondary diagonal are the elements of the partition (A') of k(p + 1) - na, and that 1P I 0 (A*) = {pi 0 (A'*) For example, when (A) _ (s'), where s and j are two positive integers, the ((p + 1 - s)'), former of which
- 2, the analysis of r. r.) into its irreducible constituents: 1'(.,)l'(a) = r(6) -t- rcu) + rcas>

while, when n = 1, this analysis is simply r(s)r. = r(b, the other two terms, r(41) and rcaa, , vanishing. This is a particular case of the following result: If (A) is a partition of any positive integer m > n having more than n nonzero parts, then (A) vanishes.

The argument proving this will be clear on considering the example (A) = (3212), n = 3. Since (oo - alt + oils - a,,e) (ho + hit + hie +

)=1


we see, on checking, in turn, the coefficients of to+a'

EH]

t8' to+a=-1

to+a,-2 = t2 and tn+)'-s = t, that

121 = t,

-- asha + u2h4 - alhs + uoho = 0 -- ash1 + u2h2 - albs + U044 + 0

a2ho - a1h1 + wok = 0

- aiho + - aoh1 = 0 Since uo = 1 1d 0, the matrix of these four linear forms in -as , a2 , all , ao is singular and the determinant of this matrix is JA}. Thus 1 A) vanishes if the number of nonzero parts of (A) is > n. For example, {ls} (12) = (161 + {21s) + {221{ (1a}1121 = {2131 +{221), 1131 112} _ 1221),

if n >- 5

if n=4

if n = 3

while 11'1 (1Y) vanishes if n C 2, the factor (1a) vanishing when n < 2. If (X) and (A') are partitions, having not more than n nonzero parts, of two positive integers, m and m', respectively,{ A) and (A') are homogeneous symmetric functions, of degrees m and m', respectively, of the characteris, zp , of the typical element, X, of the n-dimensional tic numbers, z1, unitary group, and thus 1A} (A') is a homogeneous symmetric function, of degree m + m', of z1 i - , z, . Since (A} (A') is the character of the representation r(,)r(a.) of the n-dimensional unitary group, it is a linear combination, with positive integral coefficients, of the characters of irreducible representations of the n-dimensional unitary group, and these irreducible representations must correspond to partitions, (µ), of m + m', which implies that none of these partitions contains more than m + m' nonzero parts. Thus, in order to obtain, for any value of n, the analysis, into its irreducible constituents, of the representation roa,r(%.) of the n-dimensional unitary

group, it suffices to take n = m + m'; then if n > m + m' the desired analysis is the same as that corresponding to n = m + m', while if

n <m -j- m' we merely disregard the partitions (µ) of m + m' which contain more than n nonzero parts. Furthermore, since multiplication of representations is commutative, we may assume that the number, k, of nonzero parts of (A)

is > the number, k', of nonzero parts of (A') and we do this. Since (A) vanishes if k > n, (A) ( A') vanishes if k > n, which implies that no partition, (I1), of m + m' that has less than k nonzero parts appears in the expression that furnishes (A) (A'). Indeed, E(µ) c(,)(p) vanishes only when all the co-


122

efficients cp,, are zero, by virtue of the orthogonality relations connecting the characters of irreducible representations of the n-dimensional unitary group. If a single (p) containing j < k nonzero parts appeared in the sum E(,,) c( )(p), we have merely to set n = j to see that the coefficient of the corresponding {p} is zero. Now the analysis of {A*} {A'*} is obtained by starring all the partitions which appear in the analysis of (A) {A') and since the number of nonzero parts of {A*) is A, we see that The first element of any partition (p) of m + m' which appears in the analysis, {A} (A') _ E(1 c(,,)(p}, of {A} {A'} is >- the greater of the two nuinbers A, and A1'.

The character, { x), of r(x) is the determinant of the k-dimensional matrix , k, is (htx+j, - , h,) where whose qth row-matrix, q = 1,

ig = A, + k - 1, k < n being the number of nonzero parts of (X). On denoting by t:, ,

q=1v...gk, an operator which decreases by 1 the subscript of the qth factor of a product of k h's we may express this fact as follows: { A} is the result of applying hrk the determinant of the k-dimensional matrix whose qth rowto hi,

matrix is W-1,

,

i 1). Since this determinant is the difference prod-

uct, a (t) = (, - l Z)

of the k operators E1

,

, Sk , we

have the relation

{A) = n(f)hj, ... h,k so that

{A} (m} = a(E)h,1

- - hi,thx.

On denoting by (E') the set of k + 1 operators 51 , relation

, tk+1 we have the

{(A), m) _ A(E)hr,+1 ... the subscript of the first of the k + 1 h's, for example, being A, + k = 11 + 1

insteadof A, + k - 1 = 11 as before. Now &(t') is the product of o(E) by (4 - 4+1) = (1 . Sk} (1 -+1ti1+') (1 -+lk '} (S1 - Sk+1) and so

{(A), m) _ (1 -

tt

-

(1 -

(1 - Sk+1S11) ... (1 The reciprocal of (1 - &k+/,tti')t

tt Sk+1-1

t ... hlkh.rro A(E)h,,

') 1X) {m}

(l - ttk+1tt') is

1 + h1(C ')Sk+1 + h2(S-')Sk+1 +


1 23

and so (A) (m} may be obtained by applying the operator

1 + hl(r) 4+1 + )6(r) i;x+1 +

to ((A), m}

Writing ((A), m) in the form A(') h11+1

of applying h1(r)4+1 = (E 1 + 4(l;')(ht,+2ht,+1 .

hlk+,h,,, , we see that the result - + E-')4+1 to ((h), m) is

htk+l + ht,+lh:,+2h, +1 +

+ ht,+, ...

11\1+1,A2i...,'\k,M- 11 +(A1A2+1, ...,Ak,In - 11

+... +

I'M- 11

Since every (- ) ending in a negative number vanishes the series

1+hl(E-') t+1+ may be broken off at the -'):;"k`+1 term. In particular, when m = 1, it may be broken off at the h,(r)ik+, term, and we have a result which we have already encountered: (A) (1)

where, if k = n, the last term vanishes. This result furnishes the analysis, into its irreducible constituents, of the product of any continuous irreducible representation of the n-dimensional unitary group by the self-representa-

tion of the group. Similarly, since h2(f') = Ei-1 E 2 + Ep e2 >

> ek > 0, is (-1) k(k-,>rs(E)


125

, ti, = ek , and the (a) a(a), where ei = ei - k + 1, e2 = eg - k + 2, determinant of the k-dimensional matrix formed by the (ek + 1)th, (ek_1 + 1) th, (e1 + 1) th columns of the k X co matrix whose qth (-1)kut-uts{e} row-matrix is (hq,k('), hg_k+i(E'), . ) is (E') so that the reciprocal of Ha-i(1 - 1'ap) ... (1 - E.8) is 1 + E1.>{ of (a) (e}

W), the summation being over all the partitions (e) of all positive integers. Thus we have the following rule:

(A} (A'} is the result of applying the operator 1 + Do (e} (a) { if (t') to ((A),(A')}

It follows from this rule that no partition having more than k + k' nonzero parts appears in (A} (A') . On combining this with a previously obtained result we see that the partitions of m + m' which appear in (A} (A') have not less than k and not more than k + k' nonzero parts. This implies,

on considering the associated partitions, (X*) and (A'*), to (A) and (A'), respectively, that The first elements of the various partitions, (p), of m + m' which appear in (A} (A'} are all greater than, or equal to, the greater of the two numbers Al and A1', and not greater than Al + Ai'.

The work involved in applying the rule just given is lessened by the fact that if (e) and (A) are any two partitions, having not more than n nonzero parts, of any two positive integers, then ( of () (A) is the determinant of the n-dimensional matrix obtained by reducing by e, , j = 1, , no the subscripts of the h's in the jth column of the n-dimensional matrix whose determinant is ( X1. To prove this we observe that the product of (e) (E) by t,() is the determinant of the n-dimensional matrix whose qth , row-matrix, q = It , no is Q.", .. , '"), where ei at el + n - 1, e. = e". Hence, since (A} = o() h,1, .. , h," where 11 = Ai + n - 1, - , 1" = X. , { of (1) (Af is the result of applying to h,, h," the determinant of the n-dimensional matrix whose qth row-matrix is (eo'', , In other words, {e} () (A} is the determinant of the n-dimensional matrix whose

9th row-matrix is (hi,.1 , . , ht,-a") _ (lli,_n+i_,i , htr_n+s_o2 , , hl,..), which is what we wished to prove, since (A) is the determinant of the n-dimensional matrix whose qth row-matrix is (h,,-"+i , h,. +: , hi,). It follows that { of (1) { A) vanishes if ei > Al since, then, all the elements in the first column of the n-dimensional matrix whose determinant is ( of (E) (A) are zero. Furthermore, if el - Ai , (e) (e) ( A) _ I es , - , As ,

,

A") since, when ei = A1, the first column-matrix of the 1

0 n-dimensional matrix whose determinant is ( of (f) ( Al is 0 0

.

Hence (e} (E)

128


(A) vanishes unless all the inequalities el < Al , e2 < A2, - .. , e < A. hold. In particular, ( el (E) (A) vanishes if the number of which (e) is a partition is greater than the number, in, of which (A) is a partition, and when these two numbers are equal, (e) (E) (A) vanishes unless (e) = (A), in which case (el (E) (A) is (0) = 1. Furthermore, when the number of

which (e) is a partition is <m, (el () (A) vanishes if the number of nonzero parts of (e) is greater than the number of nonzero parts of (A). When the number, k, of nonzero parts of (A) is k. Thus, in the summation r(.){ e) (8) (el (i'), we may restrict ourselves to those partitions (e) of all positive numbers which have not more than k' nonzero parts, and for which the number of which (e) is a partition is <m'. In evaluating (el (8) (A), we may take n to be k, which simplifies the calculation if the number of nonzero parts of (e) is k, for we may then replace (el by (ek) (el - ek , - - , ek_, - ek). For example, (121(8) (43) = (541. To illustrate the method we evaluate (431 (211, the first term of which is (43211. Since (1J (6) 1431 = (531 + (421, (1) 0) (211 = (21 + (12) the next set of terms is (5321 + (5312) + (4221 + (4212). Since (2)(8)(431 = (631+ 154), the term 1451 vanishing, and since (210)1211 = (11, the next group of terms is (6311 + (5411. Since {12)(8)(431 = (541 and (1210)(21) = (11,

the next group of terms reduces to the single term (541). Finally, since (211(8)(431 = (12)(11(8)(43) = (12)((531 + (42)) = (64) + (521, the last group of terms is (641 + (52). Collecting the various terms, we have 14311211 = 1641 + 16311 + (52) + 2(541)

+ (532) + {53121 + 14221 + 14212) + {4321)

and this relation furnishes the analysis, into its irreducible constituents, of

It is not hard to see that (el (8) (A) is the determinant of the n-dimensional matrix obtained by increasing by a"_ j the subscripts of the h's in the jth column of the n-dimensional matrix whose determinant is (A). In fact, (el (8) is the quotient of the determinant of then-dimensional matrix whose , 6,") by A(8), and this is the same as qth row-matrix is (8Q'',

(-I) A("-1)12 times the quotient of the determinant of the n-dimensional

matrix whose qth row-matrix is (8v'", -

-

8Q'') by O(8). Since A(E) =

and since (Al = of )h,, ... h,, (e) (6)(Al is the result of applying to ha,h),,_, ... h).-.*, the determinant of the In other n-dimensional matrix whose 9th row-matrix is (8Q ", - - , words, (e) (8) (Al is the determinant of the n-dimensional matrix whose qth row-matrix is (ham+i+.. , ... , = (hi.-"+,+." , ... , hi.+.,)


127

which is what we wished to prove. When the number, k, of nonzero parts of (A) is 0, and n is even, [A] is a compound character, being the sum of two simple characters [AJ+ and [A]- . When (X) has more than k but not more than 2k nonzero parts, we define the function [A] as follows: If the number of nonzero parts of (X) is k + j,

1 < j < k, we write (X) as a 2k-part partition, whose last k - j parts are zero, and we set 1, = A, + 2k - p, p = 1, - , 2k, rather than

1,=A,+k-p,

p= 1, ,k

as before. Then [XI is, by definition, the determinant of the 2k-dimensional matrix whose pth row-matrix is (hi,-sk+a , hi Yk + hi,-sk+s,

.. , hi,-+k+s + hi,)

Thus [AJ is a generalized character of the n-dimensional rotation group and Nye propose to express it in terms of the functions [A] for which (X) contains not more than k nonzero parts. The rules which tell us how to do this are known as the modification rules for the n-dimensional rotation group. 129

180


We treat first the case where n = 2k is even. We know that, if f(t) is the polynomial,

f (t) _ (1 - zit) .. (1 - zskt) = 1 - Qjt + a s h " + or t _ (1 + t2`) - oi(t + tom`-') + ors(t2 + t"-2) - ... + (-1)`okck in an indeterminate t and if tj , - - , to are 2k indeterminates, then the reciprocal of the product f(ti) f (t2k) is 1 + E(p) (,j} (z) (µ} (t), where the summation is over all the partitions (p), having not more than 2k nonzero parts, of all positive integers. On applying the operator ti' - Esk k to

... I'S", we ob-

{f(tj) ... f(tsk)}-', written in the form F,(3) h3i ... hi,kti' tain FI(J) h11-PI

hick-Dktj'

tSkk

= tip' ... t k11

,(f') hi', ...

lqk:.

where ji' = jj - pi , , j" = jzk - psk . Thus the result of applying the operator tf' ... Egt" to (f(tj) ... f(tgk)I-' is to multiply it by ti' . tk:r . Now (pJ(z) is the result of applying to h,',,

hn,,, , where

j = 1, ...,2k

m,=p,+2k-j,

the determinant of the 2k-dimensional matrix whose jth row-matrix is (E;k-I, Eik + E?s, ... , E? 2 + 1), and this determinant is

(ti ...

tsk)ft-i

times the determinant of the 2k-dimensional matrix whose jth row-matrix is

(1, t, + E}', (t, + tJ')2, ... , (E; + E7j')2") The determinant of this latter Vandermonde matrix is sk

P' Q

2k

t

(ED + Ep' - t4 - to') _ II (t4 - EP) (1 - EQED) Q 0, r,),, is the product of r(,,) where (µ) = (A, - X. , , An_1 - A.) has not more than n - 1 nonzero parts, by the Apth power of the determinant representation. We suppose, therefore, that (A) does not have more than 2k nonzero parts, where n = 2k + 1, when n is odd, and n = 2k, when n is even. We have seen that [A] is the result of applying to (A) the operator 1 Q (1 - EpEQ) and so (A) is the result of applying to [A] the reciprocal of this operator, which is

1 + (2)(E) + ((2) 0 (2))(E) + ((2) 0 (3))(E) + ... We shall see in the next chapter that (2) 0 (j), j = 1, 2,

, may be obtained by the following rule: Take each partition of j and double all its parts. For example:

(2) 0 (2) = (4) + (22);

(2) 0 (4) = (8) + (62) +

(2) 0 (3) = (6) + (42) + (2') (422) + (2`)

and so on. When (A) is a 1-part partition, we need only the 1-part terms of these expressions, since I p) (E) (A) vanishes when (p) has more nonzero parts

than does (A) ; indeed, this implies, since [A] = fl g (1 - EpEQ) (A), that (µ) (E) [A] vanishes when (µ) has more nonzero parts than does (A). Thus , the right-hand side ending with [0) or [1] (m) = [m] + [m - 21 + according as m is even or odd, respectively. When n = 2, the various representations of the 2-dimensional rotation group whose characters are [m], are reducible, with the single exception of the identity repre[m - 2], sentation, whose character is [0]. If j > 0, the 2-dimensional representation of the 2-dimensional rotation group whose character is U1 is the sum of two 1-dimensional representations, whose characters are [l7+ = exp ij4 and [1u_ = exp - ij4, respectively. When n = 3, the various representations of , the 3-dimensional rotation group whose characters are [m], [m - 21, are all irreducible. When (A) is a 2-part partition we need only consider the 1- and 2-part terms of the various expressions (2) 0 (j), and we may break when 2j becomes greater off the series 1 + (2) (E) + ((2) 0 (2)) (E) + than the integer m of which (A) is a partition. Thus (1$)

= [121 + (2)(E)[12J = [121


135

so that the representation of the n-dimensional rotation group, n > 4, which is induced by the antisymmetrized square of the self-representation

of the n-dimensional unitary group is the representation of the rotation group whose character is [121. When n = 3,1121 = ill, so that the antisymmetrized square of the 3-dimensional rotation group is the self-representation of the group and hence is irreducible. When n > 4, the antisymmetrized square of the n-dimensional rotation group is irreducible, since [A] is irreducible for all values of Ak when n is odd, and for Ak = 0 when n is even. The case n = 4 is exceptional, the antisymmetrized square of the 4-dimensional rotation group being reducible. It is the sum of two 3-dimensional irreducible representations whose characters are [ll+ and 1121-. (Note. The representation of the n-dimensional rotation group whose character is [l2] = [1R] = oy is the adjoint representation of the group. Indeed, if M v, , p < q, is any one of the n(n - 1) /2 characteristic matrices of the group, and if X is a typical element of the group, the coefficient of Mp, in XM,,X', when this matrix is expressed as a linear combination of the characteristic matrices of the group, is XpPX,° - X,PXJ°. Hence the character of the adjoint representation is the sum of the two-rowed principal minors of X and this is a,2. Thus we have the following result: The adjoint representation of the n-dimensional rotation group is irreducible, save when n = 4, in which case it is the sum of two irreducible 3-dimensional representations of the group. When n = 2 it is the identity representation and, when n = 3, it is the self-representation of the group.) Let, now, (X) and (A') be any two partitions, each having not more than

k nonzero parts, of any two positive integers, m and m', not necessarily different. Writing k

k

J

/

[A'] = II, (1

1

1

we have k

k

II (1 - ep54)

p 2, or if k >- 2 and n is odd and, hence, >5, the three representations of the n-dimensional rotation group whose characters are [5], [41], and [321 are irreducible, and the formula [3][2] _ [51 + 1411 + [32] furnishes the

analysis, into its irreducible constituents, of the product of the two irreducible representations of the n-dimensional rotation group whose characters are [3] and [2], respectively. When n = 4, the representations of the 4-dimensional rotation group whose characters are [411 and [32] are reducible, each being the sum of two irreducible representations of the same dimension. In this case we write 13][21 = 151 + [411+ + 1411- + [321+ = 1321-


137

and this formula furnishes the analysis, into its irreducible constituents, of the product of the irreducible representations of the 4-dimensional rotation group whose characters are [31 and [21, respectively. When k = 1, so that n = 2 or 3, we have

(1 + E(4)(el()Iel(E')) [111

[31

(1 + E Iel(E)Iel(E')) [:321 = [321 + 1211 + I11 I

so that [3][21 = [51 + [,111 + [321 + [31 + 1211 + [11. For the 3-dimensional rotation group this reduces, by virture of the modification rules, to 131121 = 111+141+[3] +121+[11

18

The Analysis of the Representations r(12) p (X) and

r

(2)

p (A) of the

n-Dimensional Unitary Group

The character (X) of the irreducible representation r(%) of the n-dimensional unitary group appears, when expressed in terms of the power sums

ai = zi -I- ... + Z. , s s = z12 -l- zs -I- ... + zas, .. . of the characteristic numbers of the typical element of the group, in th form (a)

the summation being over all sets (a) = (a, , , a,,,) of m nonnegative + ma, = m and the coeffiintegers, which are such that al + 2as + cients c(a) being nonnegative integers divided by ml. For example, {1s), = 3i (sis - 38,82 + 283)

{2). = 2( (8,s + 82),

and so on. When 8 = (8, "

, 8s , 81

. ) is replaced by ft P 83 188

84

)


1 39

we have

{a), _

(-1)atta,t...C(a)81a,82

... sm °' _ {A*),

(a)

where (X*) is the partition of m which is the associate of the partition (1) of m. For example, {ls),. = 31 (s,$ + 38(s2 + 283) = {3),

Let us denote {X), by S, and {a),. by (S*), The values of S, and (S*), at the point of the parametric space which corresponds to an n-dimensional unitary matrix whose characteristic numbers are

(zi , ...

, za1) ,

j = 1, 2, .. . ) by (8j, 82j, ) in S, and (S*), ,

are obtained by replacing (s, , 82, respectively, and we denote the results of this substitution by S; and (S*) i , respectively, so that Si = E e(a)81"82J

. 8.a3(a)

(S*)1 = Lr

sa

(a)

If the positive integer j is even all the subscripts j, 2j, , mj are even, and so (S;) *, the result of substituting 8* for s in S j , is given by the formula

(S,)* _

(-1)a,+at+...e(a)gia,

(a)

... Smf

Since

at + 2as + ... + mam = in,

a,+as -I- as -I- ... = m - (2a2 +2a3+4a4+4as-I- ...) is even when m is even and odd when m is odd, and so (-1)at+as+... _ (-1)m(-1)al+a,+... Hence

(S,)* = (-1)m(S*), On the other hand, when j is odd, pj is even when p is even and odd when p is odd, p = 1, 2, , m, so that

(S,)* _

(-1)a;ta,+...e(a)8'ai

(a)

... 8: = (8*),

If (1') is a partition of any positive integer m', not necessarily different


140

from in, ro,) ® (A') is the representation of the n-dimensional unitary group whose character is Eis> c( 's)S1°i

Sow', where

C($) al"' ...

{

S..

the summation being over all the sets of m' nonnegative integers,

(P) = (PI , ... I that satisfy the relation 01 + 2142 + ... + m'g9m - m'. Replacement of (X) by its associate partition (X) is equivalent to replacing S, by (S*), , , so that the character of the representation ro,.) ® (A') of j = 1, 2, the n-dimensional unitary group is L(e) c1e>(S*)1"' (S*),,, . When ?n is even, this is the same as Do) c($)((81) *) a' .. ((Sm ) *)"°°', which is the result of replacing s by a* in the character of r a,> ® (W). Thus we have the following result : The irreducible representations of the *-dimensional unitary group which appear in the analysis of r(),>. ® (A') are, when m is even, the associates of the irreducible representations which appear in the analysis of r&) ® (A'). On the other hand, when in is odd, (St) * = - (S*), , when j is even,

and (S1*) = (S*) i when j is odd and we have the result that: The irreducible representations of the n-dimensional unitary group which appear in the analysis of ro,>. ® (A') are, when m is odd, the associates of the irreducible representations which appear in the analysis of r(x) ®(A'*). For example, the analysis of r(2) ® (A') may be obtained from the analysis of r(1,) ® (A') by taking the associates of the representations appearing in this latter analysis and the analysis of r(3) ® (A') may be obtained from the analysis of r q>> ® (A'*) by taking the associates of the representations appearing in this latter analysis.

In order to obtain the analysis into its irreducible constituents of the representation r(p) ® (1") of the n-dimensional unitary group, we separate

one of the characteristic numbers say z1, of the typical element of the group from the remaining n - 1, 2t, - - , zR , which we denote by z'. Then (1 - z1,r t) = 1 - {12} t + ({ 12) ®(12)) t2 , where t is any

fl

1 >

> b..

Thus when in = 6, the b's are furnished by the partitions (6), (51), (42),

(321) of 6, and these correspond to the partitions (' (52'1'),

(34

10)

2) = (432x1) and (321

ras> ® (18) =

=

= (61'), (5 ?)

(34) of 12, respectively. `Thus

r(,2,s) + r(4n2,) + r(.)

From these results we read off the following analyses:

ra) ® (1`) =

r(2) ® (is) = F(412) + 1'(32) r(2) ® (15) = r(.su ;

r(x) ® (16) -

r(6,s) + r(.,) + r(4s)

r(,) ® (12) = r,3,> ;

r(as,s) + r(4=2)

and so on, the various partitions of 2m which appear in the expression that

furnishes (2) 0 (1'") being the associates of the partitions of 2m which appear in the expression that furnishes (12) ® (1,*). We obtain, in the same way, the analysis, into its irreducible constituents, of the representation r(,s) ® (m) of the n-dimensional unitary group. The

reciprocal of the product II (1 - z,,z2t), where t is any indeterminate whose modulus is ® (5) = x110) + r.a2) + x(64) + r(a21) + r(462) + r(.2=) + x(26)

and so on. It is always understood that partitions having more than n nonzero parts are discarded. For example, for representations of the 2-dimensional unimodular unitary group,

r(2) ® (5) = roo) + x(32) + rce.) = r f1o> + r(6) + r0> This relation tells us that the symmetrized 5th power of the self-representation of the 3-dimensional rotation group is the sum of the three irreducible representations of dimensions 11, 7 and 3, respectively, of this group. Having obtained the analyses of rot) ® (1'") and rat) ® (m), we ob-


144

tain the analysis of r(i1) ® (X), where (X) is a partition of m other than (1'") or (m), by a method which will be clear from the following examples. Since (121 111 = 1211 +{131, (1121 ® (12))({111® (11) _ ((121 0(21))+ (1121 ® (13)). Hence 1121 0 (21) = 12121 112) (3111 - (211 = 13211 + (2112$ + (21'1, so that

-

rue) ® (21) = mast) +

r(21,2) + r(214)

From this we read off the analysis res> ® (21) = r (53) + rc42) + r casu

Similarly,

(11)®(212)_142111+(322$+13221)+{3213$+1316)+(3161+12'121 and 1121 ® (31) _ (431)+ (32121.132211 + (32131 + 12')21+ {22141 +12161,

To obtain 1111® (22) we avail ourselves of the relation (11 1211 = (311 + (22$ + (2121. We find that (11) 0 (22) = 14221 + {3212$ + {3213$ + 12'1 + 12214$

Thus

r(it)

(31) = r1431> + r(a612) + rcas8u + F($212) + x(2$16)

+ x(2114) + r(16)

x(2,6)

(22) = r(422) + rc3116> + r.s> + r(24) +

x($114)

r(,6) ® (212) = r(4612) + rca22) + rcf2,> + r($13) + r. s) +

r(2s12)

and from these analyses we read off the analyses: res> ® (31) =

r($) + r($3) + rtasu + x(431) + r(421> + r(ash)

res> ® (22) = r(n) + r(as1) + x(41) + x(432) + rca',1>

r(2) ® (212) = r(111) + rca,> + r(a2,) + x(43) + x(4211) + x(322)

The method used above to analyze r(,2) ® (1") and x(12) ® (m) may be used to analyze r(,s) ® (1') and r(is) ® (m). For example, in order to obtain the analysis, into its irreducible constituents, of the representation r(,3> ® (1'), we observe that the power series 1 - 112)t + ((13) ® (12))e_

is the product of the following two power series: (1)

1 - (12$'z,t + ({12) ® (12))'zi t2 - (1121 0 (13))'z,313 + .. .

(2)

1-(13)'t+({13)®(12))'9-...

The Unitary and flotation Groups

145

On equating, for example, the coefficients of t2 in the two equal power series, we obtain the relation

(12} ® (12) = (212}'z12 + (12211' + (212)' + (ls}')z1 + ((1a) 0 (12))' Hence the leading term in the expression that furnishes { 12} 0 (12) is (2212} and, since (1} (i') {2212}' = {212}' + {221)', the next and last term in the expression that furnishes 1131 X (12) is (I'). Thus

so that r(is) ® (12) = x(2212) + rile) To obtain (l2) ® (m) we observe that the power series 1 + {1') t + (1131 0 (2)) t2 + - - is the product of the following two power series (12} ® (12) = (2212} + {1a)

(1)

1 + { 12)'zit + (1121 0 (2))'z12r + - . .

(2),

1 + 1131't + ((1a} 0 (2) )'t2 + - . .

On equating, for example, the coefficients of t2 in the two equal power series we obtain the relation (12} 0 (2) = ((2211 + (1e)')z12+ (122ll' + (21a}' + { 11')z1 + ((1i} X (2) )'

from which we deduce that

{12) 0 (2) = (2'} + (214} so that r(is) ® (2) = rus) + r(2e) The representations that appear in the analysis, into its irreducible constituents, of the representation r(3) ® (12) of the n-dimensional unitary group are the associates of the representations that appear in the analysis of x(12) 0 (2). Thus r(a) ® (12) = resl) + r(a2) and, similarly,

r(s) ® (2) = r(s) + r(s2) In this way we obtain the analyses: 1'02) 0 (1a) = x(3212) + r(a23) + x(2312) +

x12212) + r(3t)

r(is) ® (3) = r(as) + r(32113) + r(als) + r(221s)

and these imply the analyses:

r(3) 0 (1) = r(n) + rcaa) + resau + r ® (12)) r(a) = (r(,) 0 (21)) + (rfa) ® (la) ), we obtain the result

ra) ® (21) = F(a) + r(?a) + r(se) + r(a21) + r(se) + r(u1) +

r(eal)

146


and this implies that r(11) ® (21) = r(3221) + r(a2>>=) + r(w)

+ r(241) + r(2111) + r(ms) + r(w)

In order to obtain r(21) ® (12) and r(u) ® (2), we may start from the relation

(21) = (1)'z2 + {1)'(1)'zl + {21)'

This relation tells us that the value of (21) at the point of the paramet space corresponding to an n-dimensional unitary matrix, whose characteris s2'z14 + (s2')2Z12 + the value of 1211' at , this point. Since 82' = {2)' - { 12)', the first two terms of the expression that furnishes the value of (21) at the point in question are {42) - 1412). Continuing in this way we see that the value of (21) at the point in question is {42) - {412) - 132) + (318) + {28) - {2212). Since 1212) = (42) + {412) + (32) + 2{321) + (31') + {28) + (2212), we have istic numbers are Z12,

{21) 0 (12) = 1412) + {32) + (321) + (2212)

{21) ® (2) = (42) + {321) + {318) + (2a) and these relations furnish the analyses r(2t) ® (12) = r(41>> +

r(33) + r(rn) + r(2 I)

r(u) ® (2) = r(a) + r(m) + r(a18) + r(2.) The representations that appear in the analysis of r(41) 0 (2) are, as they must be, the associates of the representations that appear in the analysis of

r(u) ® (i8).

Bibliography

1. I. Schur, DiealgcbraischenGrundlagenderDarstellungtheorie der Gruppen, Zurich, 1936.

2. H. Weyl, The Classical Groups, Their Invariants and Representations, Princeton University Press, 1938. 3. D. B. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, Oxford, Clarendon Press, 1940. 4. F. P. Wiper, Group Theory, New York, Academic Press, 1959. 5. F. D. Murnaghan. The Theory of Group Representations, Baltimore, The Johns Hopkins Press, 1938.

6. M. J. Newell, "On the Quotients of Alternants and the Symmetric Group," Proc. Land. Math. Soc., Vol. 53, pp. 345-355, 1951. 7. M. J. Newell, "Modification Rules for the Orthogonal and Symplectic Groups," Proc. Roy. Ir. Acad., Vol. 54, pp. 153-163, 1951. 8. M. Hall, The Theory of Groups, New York, The Macmillan Company, 1959.

9. S. Bhagavantam and T. Venkatarayudu, Theory of Groups and its Application to Physical Problems, The Bangalore Press, 1951.

10. J. S. Lomont, Applications of Finite Groups, New York, Academic Press, 1959. 11. H. Boerner, Darstellungen von Gruppen, mid Berucksichtigung der Bedarfnisse der modernen Physik, Berlin, Springer-Verlag, 1955.

147

INDEX

Addition of representations, 58 Adjoint representation of the n-dimensional rotation group, 43, 135 of the n-dimensional unitary group, 43 Alternating average of a matrix, 72 Alternating function, 69 Alternating subspace, 72 Analysis of group representations, 58 Analysis of

Characteristic plane, 22 Class of a group, 18 Class factor of the element of volume, of the 2k-dimensional rotation group, 51

of the dimensional rotation group, 53, 54 of the n-dimensional unitary group, 48 Class function, 25, 68 Class of a group, 18 Class parameters, 19, 24 Class representatives, 21, 23, 24 Clebsch-Gordan formula for the 3-dimensional rotation group, 95 for the 2-dimensional unimodular unitary group, 86 for the 2-dimensional unitary group, 87 Closure of parametric space, 4, 5 Commutator of characteristic matrices, 35 Commuting relations, 35, 38, 39 Complete symmetric function, 75 Complex vector, 1 Compound character, 68 Conjugate representation, 82 Coordinates of a vector, 1

1m1 0 (2), 108 1 rrr) 0 (18), 108

121 0 (m), 109 1X1 0 (1), 115 1a1 ® (2), 123 11'J 0 (1"'), 141

121 0 (1-),142 1111 0 (m), 143 121 0 (m), 143

Anti-liermitian matrix, 34, 88 Antisymmotrie subspace, 72 Antisymmetrized power of a representation, 74 Associate partition, 115 Average of a function over a parameter group, 30 Average of a matrix over the symmetric group, 72

Determinant representation, 82 Diagonal representatives, 18 Diagonalization of a square matrix, 17 Dimension formula for the 2k-dimensional rotation group, 102, 104 for the (2k+1)-dimensional rotation group, 106 for the n-dimensional unitary group, 70 Direct power of a square matrix, 71

Base point, 29 basis, 1

Carrier space of a representation, 57 Character of a representation, 68 Characteristic directions of a linear transformation, 15 Characteristic matrices of the unitary group, 34 Characteristic numbers of a linear transformation, 16 149

150


Direct product of matrices, 59 Direct sum of matrices, 58, 63 Element of volume of a parameter group, 30 of the 2-dimensional unimodular unitary group, 37 of the 2-dimensional unitary group, 37 of the 3-dimensional rotation group, 38, 45, 46 Elementary symmetric function, 75 Euler factorization, 12 Exceptional points of the parametric space, 27

Faithful representation, 55 Generalized character, 68, 76

Half-latitude angle, 26 Hormite's Law of Reciprocity, 109 Hermitian matrix, 6, 17 Homogeneous Lorentz transformation, 90

Identity representation, 55 In-class factor of the element of volume of the 3-dimensional rotation group, 46 In-class parameters, 19, 25 Invariant directions of a linear transformation, 15 Invariant plane, 22 Irreducible constituents of a representation, 58 Irreducible representations, 58, 63, 97

Matrix group, 33 Modification rules for the rotation group, 129, 132, 133 n-dimensional P pace, 1

Normalized class factor of the element of volume of the 2k-dimensional rotation group, 52 of the (2k+1)-dimensional rotation group, 54 of the n-dimensional unitary group, 49 Normalized element of volume of a parameter group, 30 of the 2-dimensional unimodular unitary group, 38 Numerical product of two vectors, 2 Orthogonal group, 4 Orthogonal matrix, 4 Orthogonality relations, 65, 67

Paramater group, 4, 10, 27 Parametric space, 4, 8 Parametrization of the proper time-sense preserving Lorentz group, 92 Partition diagram, 115 Pauli matrices, 6 Plane unitary matrix, 8

Positive Iermitian matrix, 56 Positive square root of a positive Hermitian matrix, 56 Product of representations, 61 Proper Lorentz group, 90 Proper Lorentz matrix, 90 Proper time-sense preserving Lorentz group, 91

Kronecker power of a matrix, 71

Quaternion units, 6

Latitude angle, 5 Left-translation of a function, 31 of a group, 28 Longitude angle, 5 Lorentz group, 90 Lorentz matrix, 90

Realization of the 1-dimensional unitary group, 6 Reducible representations, 58, 63 Reflection matrix, 11 Representation of a group, 6, 43, 55 Right-translation of a parameter group, 40

Tito Unitary and Rotation Groups Rotation group, 4 Rotation matrix, 4 Scalar matrix, 63 Self-associated partition, 115 Simple character, 68 Singular points ofthe parametric space, 12 Spin representatioho, 94 Squared magnitude of a vector, 2 Standard basis, 1 Star of a matrix, 2 Symmetric product of two vectors, 2 Symmetric subspace, 72 Symmetrized power of a representation, 74

Time-sense preserving Lorentz group, 91 Lorentz matrix, 91 Triangular matrix, 15

Unimodular matrix, 3 Unimodular unitary group, 3 Unitary basis, 3 Unitary group, 3 Unitary matrix, 3 Unitary representation, 55 Unit characteristic

n X I matrix, 10 Vandermonde matrix, 69 Zero vector, 2

151

The Unitary and Rotation Groups

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Representations of the rotation and Lorentz groups and their applications

The Rotation and Lorentz Groups and Their Representations for Physicists

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Unitary symmetry and combinatorics

Quaternions and rotation sequences

Unitary Symmetry And Combinatorics

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Stellar Rotation (Cambridge Astrophysics)

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The Unitary and Rotation Groups

Unitary reflection groups

Unitary representations of topological groups

Representations of the rotation and Lorentz groups and their applications

The Rotation and Lorentz Groups and Their Representations for Physicists

Noncommutative distributions: unitary representation of gauge groups and algebras

Unitary symmetry and combinatorics

Quaternions and rotation sequences

Unitary Symmetry And Combinatorics

Unitary Representations and Harmonic Analysis

Quaternions and rotation sequences

Stellar rotation

Unitary representations and harmonic analysis

Archimedean Zeta Integrals for Unitary Groups (2006)(en)(18s)

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Rotation and Accretion Powered Pulsars

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Lie groups and compact groups

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Lie Groups and Algebraic Groups

Stellar Rotation (Cambridge Astrophysics)

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