CHARACTER THEORY OF INFINITE WREATH PRODUCTS

The representation theory of infinite wreath product groups is developed by means of the relationship between their group algebras and conjugacy classes with those of the infinite symmetric group. Further, since these groups are inductive limits of finite groups, their finite characters can be classified as limits of normalized irreducible characters of prelimit finite groups. This identification is called the “asymptotic character formula.” The K0-invariant of the group C∗-algebra is also determined.

the infinite symmetric group.An analogous reduction was found for the infinite unitary, symplectic, and orthogonal groups.See [5] for recent applications of character theory of inductive limits of classical groups to harmonic analysis and probability theory.See [10] for an application of the asymptotic character formula to the study of multiplicities of specific representations.
We fix some notation concerning Young diagrams.A Young diagram Y is a finite ideal of (Z + ) 2 relative to the usual product order structure.Let |Y | denote the number of points in the diagram; while r i (Y ) denotes the length of its ith row and r # (Y ) denotes the number of nonzero rows; c i (Y ) denots its ith column.Finally, we denote the Frobenius coordinates of Y by where h j (Y ), respectively, v j (Y ), is the (horizontal, resp., vertical) coordinate r j (Y ) − j, respectively, c j (Y ) − j, (length measured from the main diagonal).d # (Y ) denotes the length of the main diagonal of the diagram Y .Let ᐅ n denote the set of all Young diagrams with exactly n nodes, and let ᐅ = ∞ n=0 ᐅ n .An infinite Young diagram Y is an infinite ideal of (Z + ) 2 .In addition to describing an infinite diagram Y with its row and column lengths, Y can also be specified relative to a rectangular piece together with a finite diagram.Let with k + ≥ 1.Then any infinite Young diagram Y with finitely many rows and columns has the form where Y 0 is a finite (possibly empty) Young diagram and Y 0 + (k, ) means the usual translate of the Young diagram by the vector (k, ).As a convention, for a finite group G with dual space G of equivalence classes of irreducible representations, let p(π) denote the corresponding central projection in C (G) while e(π) denotes a minimal subprojection of p(π).When G is abelian, we may enumerate its dual as {ω 1 ,ω 2 ,...,ω |G| } when convenient.

Algebraic structure of K 0 (S(G))
For a finite group G, let S n (G) denote its wreath product which is the canonical semidirect product of G n = G × G × ••• × G (n factors) where S(n) acts on G n by permuting components.Write elements of S n (G) as (σ;g 1 ,...,g n ) where σ ∈ S(n) and g 1 ,...,g n ∈ G. Given a set S, let F n (S,ᐅ) denote the set of all functions f from S into ᐅ such that s∈S | f (s)| = n.
If R(S n (G)) denotes the standard additive group of representations of S n (G), then = ∞ n=0 R(S n (G)) is a commutative graded ring with multiplication given by an induction product where denote the corresponding graded commutative ring for the symmetric groups.In [16], canonical isomorphisms T ω , for each ω ∈ G, are described from 0 onto a subring (ω).Recall that a representation π ∈ R(S n (G)) lies in (ω) if and only if the restriction π to G n is a multiple of With the maps T ω , there is a natural parametrization of the irreducible representations of the wreath product S n (G) by f ∈ F n ( G,ᐅ).This is done as follows.First, identify f (ω), for ω ∈ G, with an irreducible representation of the symmetric group S(| f (ω)|).As in [16], Finally, the induction product {T ω ( f (ω)) : ω ∈ G} will produce any irreducible of the wreath product S n (G) where n = | f |.To simplify notation, we will occasionally identify f (ω) with its image T ω ( f (ω)) and treat f as the irreducible representation.
In this paper, we will only need the conjugacy classes of S n (G) when G is abelian.They are parametrized by the set h ∈ F n (G,ᐅ).In outline, this is done as follows.Given a nonnegative integer m and g ∈ G, let C m,g denote the normalized characteristic function of the primitive conjugacy class of S m (G) which consists of all elements (σ m ;g 1 ,g 2 ,...,g m ) in S m (G) (σ m is an m-cycle, g 1 ,...,g m ∈ G) such that the cycle product of g 1 ,...,g m is g (see [16] for details).Next, for a Young diagram Every conjugacy class has this form.
The algebraic structure for extends to the infinite sum of complex-valued class functions Ꮿ = ∞ n=0 CF(S n (G)), so it forms a graded commutative algebra.Further, each space CF(S n (G)) admits an inner product from the L 2 -inner product on S n (G) relative to normalized counting measure.Let Char( f ) denote the character of the irreducible f ∈ F n ( G,ᐅ) and let h ∈ F n (G,ᐅ), then Char( f ),h is the value of the character of the irreducible f at the conjugacy class h.Furthermore, the subrings (ω) are mutually orthogonal.Let Ꮿ 0 denote the corresponding algebra for the symmetric groups.
We record two basic formulas.
Branching law.Let f ∈ S n (G) and f ∈ S n+1 (G).Then f occurs in the restriction f to S n (G) if and only if there exists ω 0 ∈ G such that f (ω) = f (ω) for all ω = ω 0 and f (ω 0 ) occurs in the restriction of f (ω 0 ) to S(| f (ω 0 )|).Further, the representation f occurs in f with multiplicity dim(ω 0 ).
We now reduce the character theory of infinite wreath products to the abelian case by using the ergodic method of finding characters for locally finite groups.We briefly review this method from the theory of AF C -algebras.See [6, Chapter 1] for a general exposition.Because the group C -algebra of S(G) is an AF-algebra, it possesses an associated dynamical system (X,Γ), where X is a certain path space and Γ is a locally finite group of path permutations [11].In particular, X consists of infinite paths through the "multigraph" whose nodes are given by f ∈ S j (G), j ≥ 0, such that the two vertices f and f are connected by k edges if f ∈ S j (G), f ∈ S j+1 (G), and f ≤ f where k is the multiplicity of f in f .The finite characters are determined by the Γ-invariant and Γ-ergodic probability measures µ on X.The measure µ is determined by its values on the cylinder sets X f , where f ∈ S n (G), for some n, and X f consists of all paths that pass through the node f .The ergodic method states that the measure µ is uniquely determined by the limit where 0 is the trivial representation of S 0 (G) = {e}, f N ∈ S N (G), and Path( f , f ,G) is the set of all finite paths whose initial node is f and whose final node is f .We recognize that #Path(0, f ,G) is the dimension of the irreducible representation Let t be the associated character of the C (S(G)).Then the value of t on the minimal projection e( f ) is given by By the branching law and dimension theorem, we have the relationship Hence, the ratios of the orders of the path spaces are related as where p = | G|.Hence, the limit in (2.6) exists for the group G if and only if it exists for Z p .We sum up this discussion in the following.
Theorem 2.2.Let G be any finite group.Then the character theory of the infinite wreath product depends solely on the cardinality of G.
It would be interesting to have a version of this result for nonfinite compact groups G but it is not clear what the replacement of Z p is.For the remainder of the paper, we assume G is a finite abelian group.

Finite characters and primitive ideals
We give a brief review of the known classification of the finite characters of S(∞) together with their asymptotic character formula and the ergodic method.Let x denote two nonincreasing sequences {a j } and {b j } of nonnegative reals such that c = 1 − (a j + b j ) ≥ 0. We set where ) n are the generalized power sums.For Y ∈ ᐅ, let s Y (x) denote the corresponding Schur function given by the power series F(x,t), so s Y (x) = det[p ri(Y )−i+ j (x)].Then the corresponding finite character t x is multiplicative relative to disjoint cycles and is given by t x (n-cycle) = p n (x) and t x (e(Y )) = s Y (x), where e(Y ) is any minimal projection determined by the diagram Y ∈ S(n).Every finite character has this form [12].For further discussion of these results, see also [6,9,13,15].We call x the Thoma parameters of the character.
Recall the asymptotics for the S(∞)-characters.[16], let z m denote the normalized characteristic function of mcycles in S(m).Identify as usual σ ∈ S(m) with its image in S(m + k) where its image fixes the complementary indices to {1, ...,m} among {1, ...,m + k}.Then the character value of Y N at an m-cycle σ m is given by Char where z m denotes the normalized characteristic function of the conjugacy class of all mcycles in S(m) and z m z N−m 1 is the induction product in Ꮿ 0 .Then the limit of the normalized sequence of characters exists if and only if a N, /N → a and b N, /N → b such that (a + b ) < ∞.This can be restated relative to the generalized power sums given by the generating function This framework can be rephrased in terms of the ergodic method.By [6], we have the asymptotic estimate where s Y * 0 (x N ) is the Schur function relative to F N (x N ,t).Hence, for any finite character t with Thoma parameters x, there exists a sequence By means of the asymptotic character formula and the ergodic method, we will be able to classify all the finite characters of S(G).
and whenever q ω > 0, the following limits exist for all j ≥ 1: (3.5) Proof.Because of the multiplicative nature of finite characters [7,13], it is enough to establish the limits of the normalized characters on the primitive conjugacy classes which play the role of n-cycles in the infinite symmetric group.Let f ∈ F N ( G,ᐅ) = S N (G).We wish to find the character value Char( f ) at the conjugacy class C h which is nothing more than the inner product Char( f N ),C h .We will be using the formalism of [16].By [16,Section 7], we have for a primitive conjugacy class C m,g that where g ∈ G and m is a nonnegative integer.With the usual embedding of S m (G) into S N (G), we analyze the asymptotic behavior of the inner product where the sum is over all nonnegative integers n 1 ,...,n |G| that sum to N − m.
By orthogonality, the only nonzero contributions of C m,g (C 1,e ) N−m in the inner product occur when for some j 0 , we have m + n j0 = | f (ω j0 )| and for j = j o , n j = | f (ω j )|.Again we emphasize that the products here are induction products in Ꮿ.For convenience, set n i = | f (ω i )|, and let A(N − m, j) denote the multinomial coefficient N−m m1,...,m|G| where m i = n i , for i = j, and m j = n j − m.Further, set M j to be the monomial product in Ꮿ: We also need the elementary asymptotic estimate for multinomial coefficients where p, n 0 , k 1 ,...,k p will be treated as fixed: Hence, the ratio N−n0 K1−k1,...,Kp−kp / N K1,...,Kp has a limit if and only if for each j = 1,..., p, the sequence {K j /N} converges with limit q j ; hence the limit of the quotient is q kj j .We now have the simplifications where we used the fact that Char( is the value of the character of the irreducible representation f N (ω i ) of S(| f N (ω i )|) at the identity which reduces to the dimension of the representation.
We conclude that the normalized character value is Since the character value on S(N), can be handled with the asymptotic character formula for S(∞), we will give now a full asymptotic expansion for the normalized character value: where then this term has no contribution to the sum.So, by passing to a subsequence, if necessary, assume | f N (ω i )|/N → q ω > 0. By [13, Lemma 2], (3.13) possesses a limit for such ω if and only if both a ( f (ω)) and b ( f (ω)) possess limits, say a (ω) and b (ω), respectively, for all , together with the normalization a (ω) + b (ω) = 1 − c(ω) ≥ 0. In particular, this term has contribution q ω p m (z(ω)) to the sum.Since the characters ω are a basis for functions over G, we find that distinct limits give distinct finite characters.
Theorem 3.2.The finite characters t of S(G) are parametrized by (q ω ,x(ω) : ω ∈ G) where q ω ≥ 0 with q ω = 1 and x(ω) = (a (ω),b (ω)) are Thoma parameters for each ω.On the primitive conjugacy class with g ∈ G and m a nonnegative integer, the value of t is and, on the minimal projection e( f ) with In particular, q ω = t(e(ω)) where e(ω) is the minimal projection corresponding to ω ∈ S 1 (G) in C (S 1 (G)).
Proof.The evaluation of the finite character t on a primitive conjugacy class follows from the proof of the previous theorem.
To obtain the values of the finite character t on minimal projections in C (G), we use the ergodic method.For f * 0 ∈ S n (G), we find Hence, the value of finite character t at the minimal projection e( f * 0 ) is given by the limit: In particular, where Hence, q ω has the desired interpretation.
We will call the parameters (q ω ,x(ω) : ω ∈ G) from Theorem 3.2 for a finite character t of S(G) its invariant parameters.
The method in [13] or [3] can be used to classify the primitive ideals of C (S(G)) by means of the branching law.For each ω ∈ G, we let Y (ω) be an infinite Young diagram described in Section 1, so Ᏽ k(ω), (ω) + Y 0 (ω), where k(ω) is the number of infinite rows of Y (ω) and (ω) is the number of infinite columns of Y (ω).Note: we allow either k(ω) or (ω) to be ∞.In either case, we set Y 0 (ω) = ∅.
A primitive ideal J is determined by the G-tuple: Hence, the minimal projection e( f ) ∈ J(k, ,Y 0 ), where f ∈ S n (G), if and only if f (ω) is not a subset of Ᏽ k(ω), (ω) + Y 0 (ω), for some ω ∈ G.
(2) The parameters for the finite characters of S(G) that come from the quotient group n=1 G n are determined by q ω1 = 1 where ω 1 is the constant character 1 on G.

Infinite characters
The aim of this section is to classify the infinite characters of S(G) using Riesz ring techniques.We recall some concepts from Riesz group and ring theory.First, an ordered group G is a Riesz group provided given any elements a i ,b j ∈ G, i, j = 1,2, there exists an element c ∈ G with a i ≤ c ≤ b j for any choice i, j = 1,2.For any unital AF-algebra, its K 0 -group is a Riesz group.A Riesz group is a Riesz ring provided it has a multiplication that is consistent with the order structure.
If J is the kernel of a character t, whether finite or infinite, then t corresponds to a faithful character of the primitive quotient A = C (S(G))/J.Further, with the induction product described in Section 1, K 0 (A) is a commutative ring.In particular, K 0 (A) for a primitive quotient A is a Riesz ring.
Let H denote a Riesz group with nonzero positive elements a and b.We say that a is infinitely small relative to b if na ≤ b for all n ∈ Z + .If a is infinitely small, then a must lie in the kernel of every state of H.In particular, if H is a unital Riesz ring with a nonzero infinitely small element, then H can have no faithful finite extremal states.
We need the following theorem found in Antony Wassermann's unpublished University of Pennsylvania Ph.D. thesis.Although the thesis was widely circulated, it was never published.A version of this result with a proof by Wassermann appears in an appendix to [3].It is routine to check that the primitive quotients by the ideals J = J(k, ,Y 0 ) where Y 0 (ω) = ∅, for all ω ∈ G, have no positive zero divisors.In another study [4], we found for several inductive limit groups that a primitive ideal J is integral (i.e., the K 0 -ring of the primitive quotient is a commutative ring with no positive zero divisors) if and only if J is the kernel of a finite factor representation.
Throughout this section, we let J = J(k, ,Y 0 ) where for some ω 0 ∈ G, Y 0 (ω 0 ) = ∅ where we use the notation introduced in (1.3).Recall that k denotes the number of infinite rows and the number of infinite columns, so the shape of such an infinite Young diagram is an infinite "L."The finite diagram Y 0 denotes the remaining contribution to the shape.We will write f ⊂ Y + I(k, ) if f (ω) ⊂ Y (ω) + I(k(ω), (ω)) for all ω.We use the same convention for other standard set operations.
We call a primitive ideal integral if the condition that a,b ∈ K 0 (S(G)/I) are nonzero and positive implies that their product ab is nonzero.We let J a denote the smallest integral primitive ideal that contains J, so that J a = J(k, ) (where we omit Y 0 here since they are all equal to ∅).
Proposition 4.2.The primitive quotient A=C (S(G))/J contains a nonzero ideal B which is stably isomorphic to an ideal B a of A a = C (S(G))/J a .
Proof.Consider the projections e( f ), where f ∈ S n (G), in A satisfy the condition We associate to ).Next consider e( f a ) as a projection in A a .By the branching law, the ideal B in A generated by all projections e( f ) satisfying (4.1) is stably isomorphic to the ideal B a in A a generated by the projections e( f 0 ), where f 0 ⊃ {(i, j) : i ≤ k(ω)and j ≤ (ω)}, which are rectangular diagrams.
We need to strengthen Proposition 4.2 in order to show that B is the norm closure of the ideal of definition of any faithful character of A. We study this question in terms of multiplication of K 0 (A).
Proof.For the first identity, it suffices to show that e( f ) • e( f ) = 0, where f is chosen, so f (ω) is the smallest diagram that contains Y 0 (ω) + (k(ω), (ω)).Without loss of generality, we may assume G = {e}; that is, we check only the S(∞) case.
We set Let p be a term in the decomposition of f • f , which is described by the Littlewood-Richardson rule [8], such that p ⊂ J(k, ,Y 0 ).According to the Littlewood-Richardson rule, the terms p are determined by the proper insertion of integers into the Young diagram.Let a i be the number of i's placed in the ith row of p.By hypothesis, a 1 ≥ 1; moreover, each a i ≥ 1, 1 ≤ i ≤ , since r j (p) ≤ k, j > .Thus, ( + 1) entries must be placed in the first k distinct columns of p.This is impossible, so e( f ) • e( f ) = 0 in K 0 (A).The second identity follows by the same method by applying the Littlewood-Richardson rule.
Lemma 4.4.Let R be a commutative unital Riesz ring, with identity e. Suppose S is an ideal of R with order unit f , with f ≤ e.If S has trivial multiplication, then for any a ∈ R + \ S + with a • f = 0, where a • (e − f ) n = 0.In particular, f is infinitely small relative to e. Proof.(1) Recall that there is a natural bijection between the faithful characters of any separable primitive C -algebra Ꮽ with any closed nonzero ideal where the from Ꮽ to is given by restriction while the extension T of a faithful character τ of to Ꮽ is given by where x ∈ Ꮽ + (see [7, page 28]).Hence, there is a natural bijection between the faithful characters of A and A a , where A a is given in Proposition 4.2.Furthermore, A a has no faithful infinite characters by Theorem 4.1, so all the faithful characters of A are classified by the finite characters of A a .
(2) By Proposition 4.3, K 0 (A) has a nontrivial multiplication while the multiplication restricted to K 0 (B) is trivial.By Lemma 4.4, any faithful character T of A must be infinite; in fact, T(e( f )) = +∞ for any f ⊂ I(k, ), while T(e( f )) < ∞ for f \ I(k, ) = Y 0 + (k, ).We conclude that the norm closure of domain of T must contain the ideal B. To show equality requires further calculation.
Let Y * denote a diagram obtained from Y 0 by deleting exactly one node.Next, for a diagram Y 1 , we introduce the notation Rep(n,Y 1 ) for the set of all f ∈ S n (G) for some n such that f \ I(k, ) = Y 1 + (k, ).So, for f ∈ Rep(n,Y 0 ), e( f ) ∈ B. We will also identify representations with their images in K 0 (A).As a consequence, if π 1 is a representation of S m (G) and π 2 of S n (G), then their induction product π 1 • π 2 = Ind Sm+n (G)  Sm(G)×Sn(G) (π 1 × π 2 ) reduces to the image in K 0 (A) of the subrepresentation of their induction product whose irreducible components f satisfy f \ I(k, ) ⊂ Y 0 .Let 1 A denote the multiplicative identity for K 0 (A) which is equivalent to the induction product with respect to the left regular representation of S 1 (G) = G.
Given any representation π of S m (G), let sub(π,Y * ) denote the subrepresentation of π whose irreducible components f satisfy f \ I(k, ) = Y * .Then for f * ∈ Rep(n,Y 1 ), let f * ,1 = sub( f * ,Y * ) and f 0 = sub( f * ,Y 0 ).By the branching rule, we find that sub( f * ,1 • 1 A ,Y 0 ) is equivalent to f 0 • 1 A .In particular, in K 0 (A), working with minimal projections, we have e( f * ) ≥ 2e( f 0 ).We obtain e( f * ) ≥ ke( f 0 ) in K 0 (A) for all k, by iterating this construction.We conclude that T(e( f * )) = ∞. ) The order structure on K 0 (S(G)) can be determined very analogously as for S(∞) (see [7] for the infinite symmetric group case).Recall that both finite and infinite characters are needed to determine the positivity of a nonzero element x in a Riesz group: x is positive if and only if x is contained in a minimal order ideal I such that (i) for all finite character t whose restriction to I is nonzero, t(x) > 0, and (ii) for all infinite characters T whose restriction to I is finite and nonzero, T(x) > 0.
Following [15], we let x = {c f e( f ) : the set of all ω ∈ G such that core(x)(ω) ⊂ I(k(ω), (ω)) is nonempty.For ω 0 ∈ S, choose Y min (ω 0 ) to be minimal relative to inclusion among all solutions of the set identity where Y is from supp(x(ω 0 )).Let (ω 0 ) denote the set of all such diagrams Y min (ω 0 ).(1) If core(x) ⊂ I(k, ) and t is a finite character whose kernel is J(k, ), then t(x) > 0.
We conclude with several comments and examples.
(1) Let J = J(k, ,Y 0 ) ∈ Prim(S(G)) such that for each ω ∈ G, k(ω) + (ω) = 1, and Y 0 (ω) = ∅ with primitive quotient A. Then the Bratteli diagram of A is a multidimensional version of the Pascal triangle.For the invariant parameters for these characters, we find a 1 (ω) = k(ω) and b 1 (ω) = (ω) and no constraint for q ω .See [7] for a discussion of its K-theory.It is also routine to describe its general factor representation of A from the results in [1].
(2) A primitive ideal J = J(k, ,Y 0 ) of C (S(G)) is type I ∞ if and only if there exists ω 0 ∈ G such that (1) for all ω = ω 0 , k(ω) = (ω) = ∞, (2) for ω 0 , k(ω 0 ) + (ω 0 ) = 1 (i.e., there is a single infinite row or column), and (3) Y 0 (ω 0 ) = ∅.Note if Y 0 (ω 0 ) = ∅, the ideal J is type I but finite.(3) The primitive ideal space of a locally finite group H cannot always be parametrized by the kernels of traceable factor representations unlike the examples in this paper and for connected Lie groups.Using the results of [2], this holds even if K 0 (H) admits a nontrivial ring structure.Let H denote the weak infinite product of S(∞), so C (H) is the infinite tensor product of the single C (S(∞)).In the notation of [2], let J be the primitive ideal of C (H) such that J = (J i ) where each J i is the kernel of the same traceable representation irreducible infinitedimensional representation of C (S(∞)).It is easy to see that such representations exist.By [2, Theorem 6.5], J cannot be the kernel of any traceable factor representation.(4) In [14], it is commented that in the product space realization of the finite factor representations of S(∞), the representation automatically becomes a type III factor representation when the product measure no longer has constant factors.This is not true in general.For S(∞), certain factor representations with primitive kernels consisting of two infinite rows or an infinite row and column provide examples.More general cases are combinatorially more difficult [1].
equals the multiplicity of the representation f in the restriction f to S | f | (G) and will be denoted by dim G ( f \ f ) or dim( f \ f ), again if the choice of G is clear.For the symmetric group, this multiplicity dim(Y \ Y ), where Y ,Y ∈ ᐅ and Y ⊂ Y , is also known as the dimension of the skew-diagram formed by the set-difference of the Young diagrams Y and Y .

Theorem 4 . 1 .
If R is a Riesz ring with no positive zero divisors, then R has no faithful infinite characters.

Theorem 4 . 5 .
Let J = J(k, ,Y 0 ) be a primitive ideal with |Y 0 | ≥ 1.Then the primitive quotient A = C (S(G))/J contains an ideal B such that (1) the faithful characters of A are all infinite and are in natural bijection with the faithful finite characters of C (S(G))/J a ; (2) the norm closure of the ideal of definition of any faithful character of A is exactly B.

SinceCorollary 4 . 6 .
A is a primitive C -algebra, any e( f ) ∈ Rep(Y 2 ), where Y 2 is nonempty, must contain a subprojection of the form e( f * ), where f * ∈ Rep(Y * ), for some Y * .Hence, the norm closure of the domain of T is exactly B. The natural map from the space of quasiequivalence classes of traceable factor representations of S(G) to Prim(S(G)) is surjective.Corollary 4.7.Let T be an infinite character on C (S(G)) with kernel J(k, ,Y 0 ) and |Y 0 | ≥ 1.Then there is a finite character T fin with kernel J(k, ) such that

Theorem 4 . 8 .
A nonzero element z = {c f e( f ) : f ∈ S n (G)} of K 0 (S(G)) is positive if and only if the following hold.