Spherical functions on wreath products

Journal of Pure and Applied Algebra 10 (1977) 127-134 @ North-Holland Publishing Comp;any

John R. DURBIN Department

of Mathematics,

The University of Texas, Austin, Texas 78712, U.S.A.

Communicated by H. Bass Received 3 February 1976

1, Introduction Group representation theory is now a standard tool in the study of special functions; see Miller [8], Travis [9], or Vilenkin [ll], for example. In this paper this theory is used to study, for any compact Abelian group A, those functions on finite direct powers A x . . x A that are symmetric with respect to any group of permutations of the variables. The appropriate representations are the class-81 representations of wreath products, A wr B, of A with finite permutation groups B; these representations are described in Section 2. By suitable choice of basis for the representation space, the zonal and associated spherical functions on A wr B relative to R can be determined. This is done in Section 3, and leads to Theorem 3, pertaining to symmetric functions. Connections with other work, including applications, are indicated in Section 4. Only algebraic tools are employed, and thus, in particular, if A is assumed finite the results fall directly in the context of spherical functions on finite groups [9]; we completely avoid measure-theoretic considerations, as used in [2] for example. l

2. Class 1 representations of wreat We consider wreat products of the formA wrB where A is locally compact Abelian and B is a subgroup of the group S, of all permutations of (1,. . ., n}. Let F=A”=Ax... X A (direct power) denote the base group of the wreath product, and use the hybrid notation fn=(f E F and 7~E B) for a typical element of A wr B ; thus f : (1,. . ., PI}+ 14 is a function, and rr E B is a permutation. The rule of ltiplication is (f?r)(gp)= fg’kp, where and their irreducible repres

tations will be over the complex field. The irreducible representations (characters) 127

of the base group F are tensor

I28

J. R. Durbin / Spherical functions on wreath products

products of those of its component subgroups, and the group B acts on such a representation x = x1 @ * 8 xn (each xi E A) by x”(f) = x(f”-‘). The representation x * is called a B-conjugate of x (W E B). The stabilityfactor of x is the subgroup consisting of those m E B such that xrr = x. Assume (1,. . ., n} partitioned into t disjoint sets, Ut, . . ., U,, with 1 Ui I= ni. Assume ~1,. . ., xr to be inequivalent irreducible representations of A, and assume x E fi to be constructed by assigning xi to each component indexed by a number from Ui (1 ZZi S t). If we let Snl denote the wgbgroup of Sn consisting of all permutations which fix every element outside Ui, then it follows that the stability factor of x is B,: = B n (S”, x X S”,).Let p be an irreducible representation of Bx, and define xv on FB, by (x(p)&) = x(f)&r). Then x(p is a representation of FB,, and the induced representation xg f (A wr B) is an irreducible representation of A wr B. Notice that xrp can be thought of as x @cp.Ye can now characterize the set of irreducible representations of A wr B. l

’

l

l

l

l

Thewem A ([3], [6] j. A complete set of inequivalentirreducible representationsof A wr B is obtained ~(ry forming allxrg 7 (A wr B), wherex runs over a complete set of irreducible but not -conjugate representationsof F, and for each x, Q runs over a complete set of inequivalent irreducible representationsof Bx. In order to determine spherical functions on A wrB with respect to the (large) subgroup B, we mug’; determine those irreducible representations of A wr B that are of class 1 with respect to B. Ref [ 111 contains all that we shall require regarding spherical functions; ref. [9] is relevant for finite groups, and [5, Chap. VII] contains snore general result!. We say that an irreducible representation + of A wrB is of class 1 relative to B if i($ 4 B, 1) = 1, where 1 denotes the l-representation of B, and i(+ 1 B, I), the intertwining number of ~5 i B and 1, is the number of times 1 is contained in rC,i B. Theorem 1. A representation$ = XQ f (A wr B), ( grnstructedas in Theorem A, is of class 1 with respect to B iff Q is the 1-representaton of B,. roof. Making use of standard theorems on intertwining numbers ([l, Chap. VII], [5, Chap. VI; 7]), we can write

= c’((X@Q)

t

(A WrB)),

where the sum is over the set of (FBx, B)-double cosets 9 of

as representations

of

.I.R. Durbin / Spherical functions on wreath products

129

(fl m)FB, cfi n-r)-’n (fim)B cfi&,

and i(x 8 c;o,lB, 9) depends only on the double coset to which (f&J-‘(f2b2)belongs; and by definition X 8 ~@~)(fv)= (~~c~)((firr*)-'(fn)(fi~,)) and 1i?2)(f7r)= ~B((fzrrz)-'(f~)(f27T2)). There is only one (FS,, B)-double coset 9 in this case, and we have the equation FB, n B = Bx. The theorem now follows from the relation i((x@di

Bx9 1B

4 &)=

{i

if cp = l-representation if +J# l-representation

of B,, of B,.

3. Spherical functions Assume X to be an irreducible representation of F, tp to be the d-representation of B,, and let ~5= Xq t (A. wrB). Then it follows from Theorem 1 that there is, in the space of $, precisely one normalized vcctc r that is invariant under (I, 4 B. In order to determine the spherical functions of $ relative to B we shall examine the matrices for (F,relative to a basis containing that vector. Let Bp,, . . ., B,n;l be a complete collection of right cosets of B, in B. Then [A wr B : FB,] = d, and d is also the degree of $. Specifically, because 9 is‘ the l-representation, the matrices of the induced representation $ are d .X d with ij-entry of $(fw) given by (3.1)

e(fr)ij =[~(TifTTF')zXV’)

if ;TT~?T$E B,, otherwise.

and shall construct an We shall identify +!j with this matrix representation, equivalent matrix representation, to be denoted by #‘. It can be seen from (3.1) that the matrices for @ ,/,B are permutation matrices. It is clear, then, that ul: = (1, 1, . . ., 1) is invariant under 31 J B. For 2 G i G d, let Ui denote the d-tuple containing 1 in the ith position and O’s elsewhere. Then {VI, . . ., vd} is a basis for the space of #, and application of the Gram-Schmidt process to this basis yields an orthonormal set {el, . . ., ed), with el = (l/d& tr,, and, for k 32, ek = (e:, . . ., e$), where er=

--l/[(d-k+2)(d-,k+l)]$,

eP=O,

m

=l,

k+l,

k+2 ,..., d,

m =2,3 ,..., k-l,

e?=(d-k+l)f[(d-k+2)(d-k+l)]$

m=k.

We examine the matrices for + relative to {el,. . ., ed). Thus we let C = (c,) denote and cij=ej(lSiGd, 2GjGd). We must the matrix with cil - l/d! (l
3.R. Durbin / Spherical functions on wreath products

130

and rs -entry 1

-

[(d-s+2)(d-s+l)ls

t,l + t,,(d - s +

1) - 2 /=s+l

tri 3

(lersd,2eed). For spherical functions we require only the first row and first column of Tedious but straightforward computation shows that (3 .2)

Ull

(3. 3)

UIS

= [d(d

X

- s +

$ tjl + i

Uml

tjs(d -

-.

1’1

(3.4)

1 2)(d - s + l)]” S +

1)~ i

j==l

i=l

e

j=s+l

tij 1

+‘(f?r).

(1~ s s d),

1

=[did

- m +2)(d

-f:

-

tlj+(d-‘m

m + l)]! +l)i

tmjj-1

j=l

Ucm

sd)*

We are now is a F;sc3sitionto write spherical functions. Again, use is made of basic results that can be found in [ll, Chap. I]. The spherical functions are given the ~‘cfn)ll, and they are constant on two-sided cosets Bfd3.Thus any such function is determined by its values on F, and, moreover, e’(f),, = @‘QI& if f” = g for some n E B. Associated spherical functions of one kind are given by the I+V(~?T)*~ (1 s k s d), and they are constant on right cosets Bfm For such functions if f B = gp,and so these functions are alsa determined by their @‘(.f+ = $‘(g& values on E Associated spherical functions of the other kind are given by the #‘Vn)m 1 (1 G m s d), and they are constant on left cosets fwB, whence $‘cfn) = #‘(gp) if f = g. Again, values are determined on F. Application of (3.1) and (3.2) yields

Application

(3.6)

of (3.1) and (3.3) yields 1

e’Cfrr)lk

= [d(d X

Application

- k +2)(d

- k + l)lf

- x(f nwi’)+ (d - k + 1)~ (fwwi’) -

of (3.1) and (3.

(k > 1).

J.R. Durbin / Spherical functions on wreath products

(3.7)

~yf?r)ln1=

131

1 [d(d-m+2)(d-m+I)]j

x [ -x(f”i’)+(d

- m + l)x(fWG’)- jz$+,x(f”7’)]

(m > 1).

Comparison of (3.6) and (3.7), in light of the remitirks preceding (3.9, shows that the functions in (3.6) are not in any essential way different from those in (3.7), and so we retain only th se in (3.7). The next theorem summarizes what we have done. Theorem 2. The zonal spherical functions on A wr B with respect to B are given by (3.9, where x, which determines $, runs over a complete set of irreducible but not B-conjugate representations of F; these functiors are determined by their values on F. In the same way, (3.7) determines associated spherical functions on .A wr B with respect to B.

We now concentrate on the zonal spherical functions for compact A. Because $ is determined by x, we write x’ in place of $ ‘, and we drop the subscripts 11. A function h on F will be said to be symmetric with respect to B if h (f”) = h(f) for all f E F, w E B. If notation is collected and [II, p. 551 is applied, the following theorem is obtained. Here df denotes normalized Haar measure on F, and a further normalizing factor (d 1B I)! has been introduced to account for the degree of x and because x’ is being considered as a function on F rather than on A wrB. l

Theorem 3.

Let A denote a compact Abelian group, let F denote the direct power A n, and let B denote a subgroup of S,,. For each irreducible representation jy of F, let B, denote the stability factor of x in B (Section 2), let (ml,. . ., n;l) be right coset representatives of B, in B, and define x’ on F by

5

x’(f) = (IB t/d)!i=l xU’9 Then, as x runs over a complete set of irreducible but not -conjugate representatives of F (Section 2), the system of functions x’ yields an orthonormal basis for the space of square-integrable complex -valued functions on F that are symmetric with respect to B. Specifically, for each such symmetric function h on F,

where

with integration taken over the space of

132 ~ 4.

J.R,Durbin/Sphericalfunctionsott wreathproducts

Example

The simplest example of Theorem 3 arises by taking A = Z2 (cyclic of order two) and B = Sn. An appropriate set of x’s is given by {x0,. . ., xn}, where xk E fi has k non-l factors (0 G k 6 n). Here B,, = Sk x SnMkand d = is, : Bxk ] is the binomial coefficient C(n, k). Two functions in F are B-conjugate iff they have the same number of non-trivial components, and so the set of B-symmetric functions can be identified with the set (0,1, . . ., n}. Viewed in this way, with multiplicities and the factor 2-” for Haar measure on F taken into account, the orthonormality relations in Theorem 3 can be written

Thus the x I;are the Krawtchouk polynomials on a symmetric binomial distribution. (Compare [4,10.241;, where different normalizing factors are used.) Vere-Jones [lo] has shown that w th A cyclic of order k this construction yields generalized Krawtchouk polyn,omials orthogonal on a symmetric multinomial distribution. This application is also (discussed in [2], where measure-theoretic methods are used and a less explicit version of Theorem 3 is referred to. By taking A cyclic and B something other than Sn, these ideas yield orthogonal polynomials on modifications of the multinomial distribu&ion. If A is not AbtZian, the relevant algebra of bi-invariant functions is not commutative; thus A must be assumed Abelian in Theorem 3. (The obstacle here is related to Theorem 2 of [9], for example.) However, in this (A compact non-Abelian) case, the Peter-Weyl Theorem and the fact that irreducible representations of direct products are given by tensor products can be applied to show directly that an averaging process over coordinate functions, as suggested by Theorem 3, once more produces a basis for the corresponding space of B-symmetric functions on A “.

References C.W.. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras (Interscience, New York-London, 1962). C.F. Dunk1 and D.E. Ramirez, Krawtchouk polynomials and the symmetrization of hypergroups, SIAM J. Math. Anal. 5 (1974) 351-366. J.R. Durbin, On locally compact wreath products, Pacific .I, Math. 57 (1975) 99-107. A. Eirdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Vol. II (McGraw-Hill, New York, 1953). S.A. Gaal, Linear Analysis and Representation Theory (Springer-Verlag, New York-Heidelberg-Berlin, 1973). A. Kerber, Representation of Permutation Groups 1; Lecture Notes in Mathematics, Vol. 240 (Springer-Verlag, Berlin-Heidelberg-yew York, 1971). G.W. Mackey, On induced representations of groups, Amer. J. Math. 73 (1951) 576-592. ler, Lie Theory and Special Functions (Academic Press, New York, 1968). Travis, Spherical functions on finite groups, J. Algebra 29 (1974) 65-76.

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133

[lo] D. Vere-Jones, Finite bivariate distributions and semigroups of nonnegative matrices, Quart. J. cB Math. Oxford (2), 22 (1971) 247-270. [l l] N. Ja. Vilenkin, Special Functions and the Theory of Group Representations, Vol. 22, Translations of Math. Monographs (Amer. Math. Sot., Providence, 1968).