A FAMILY OF FINITE GELFAND PAIRS ASSOCIATED WITH WREATH PRODUCTS

A FAMILY OF FINITE GELFAND PAIRS ASSOCIATED WITH WREATH PRODUCTS CHAL BENSON AND GAIL RATCLIFF Abstract. Consider the wreath product G n = Γ n S n of a finite group Γ with the symmetric group S n. Letting n denote the diagonal in Γ n the direct product K n = n S n forms a subgroup. In case Γ is abelian (G n, K n) forms a Gelfand pair with relevance to the study of paring functions. For Γ nonabelian it was suggested by Kürşat Aer and Mahir Bilen Can that (G n, K n) must fail to be a Gelfand pair for n sufficiently large. We prove here that this is indeed the case: for Γ non-abelian there is some integer 2 < N(Γ) Γ for which (K n, G n) is a Gelfand pair for all n < N(Γ) but (K n, G n) fails to be a Gelfand pair for all n N(Γ). For dihedral groups Γ = D p with p an odd prime we prove that N(Γ) = 6.. Introduction Gelfand pairs are fundamental to the study of harmonic analysis on topological groups. In the context of finite groups the definition is as follows. We denote by L(G) the space of complex-valued functions on a finite group G. This is an algebra under the convolution product Given a subgroup K G, the set f g(x) = y G f(xy )g(y). L(K\G/K) = {f L(G) : f( x 2 ) = f(x), 2 K} of K-bi-invariant functions on G forms a subalgebra of L(G). One calls (G, K) a Gelfand pair when L(K\G/K) is commutative. This condition is equivalent to each of the following. The left quasi-regular representation ind G K ( K) of G in L(G/K) is multiplicity free. For each irreducible representation (π, V ) of G the space V K of K-fixed vectors in V has dimension dim(v K ). Irreducible representations of G which occur in L(G/K) are called K-spherical. These are precisely those admitting non-zero K-fixed vectors. We refer the reader to [CSST08], [Mac95, Chapter VII] or [Ter99] for proofs of these equivalences as well as general bacground concerning Gelfand pairs and their applications in the finite groups setting. Given a finite group Γ the symmetric group S n acts by automorphisms on the cartesian product Γ n of n copies of Γ via σ (x,..., x n ) = (x σ(),..., x σ(n) ). 2000 Mathematics Subect Classification. Primary 20C5; Secondary 20E22. Key words and phrases. finite Gelfand pair, wreath product.

2 C. BENSON AND G. RATCLIFF The resulting semi-direct product G n := Γ n S n is the wreath product of Γ with S n, sometimes written as Γ S n. The representation theory of such wreath products is discussed in [Mac95, Chapter I Appendix B] and, in greater generality, in [CSST06, CSST4]. Let L(Γ n ) Sn = {f L(Γ n ) : f(σ x) = f(x) σ S n, x Γ n } denote the space of complex valued functions on Γ n invariant under the action of S n. This is a convolution subalgebra of L(Γ n ) and routine calculations show that the map (.) Φ : L(S n \G n /S n ) L(Γ n ) Sn, Φ(f)(x) = n!f(x, e) (e S n the identity) is an isomorphism of convolution algebras. In particular (G n, S n ) is a Gelfand pair if and only if L(Γ n ) Sn is commutative. This is certainly the case whenever Γ is abelian. On the other hand for Γ non-abelian choose points x, y Γ with xy yx and let x, y Γ n be the points x = (x,..., x), y = (y,..., y). The characteristic functions δ x, δ y for these points belong to L(Γ n ) Sn and we have δ x δ y = δ xy δ yx = δ y δ x. Thus (G n, S n ) is a Gelfand pair if and only if Γ is abelian. The diagonal subgroup in Γ n, n := {(x,..., x) : x Γ}, played a role in the preceding discussion. The S n -action preserves n Γ n and is trivial on n. Thus the direct product ( Kn := n S n ) = Γ Sn is a subgroup of G n = Γ n S n and we consider the pair (G n, K n ). Restricting the map Φ given in (.) to L(K n \G n /K n ) L(S n \G n /S n ) produces an algebra isomorphism onto (.2) A n (Γ) := L( n \Γ n / n ) L(Γ n ) Sn, the algebra of functions Γ n C which are both n -bi-invariant and S n -invariant. Thus if either (G n, S n ) or (Γ n, n ) is a Gelfand pair then so is (G n, K n ). It follows in particular that (G n, K n ) is a Gelfand pair for Γ abelian and (G n, K n ) is a Gelfand pair for n = 2. The latter point follows from the well-nown fact that (Γ Γ, 2 ) is a Gelfand pair [Mac95, VII-, Example 9]. For cyclic groups Γ the resulting Gelfand pairs (G n, K n ) arise in combinatorics in connection with paring functions [AC2]. This fact motivates interest in pairs (G n, K n ) for other finite groups Γ. It is suggested in [AC2] that for Γ non-abelian (G n, K n ) will fail to be a Gelfand pair for n sufficiently large. The following theorems show that this is indeed the case. These are our main results. Theorem.. If (G n+, K n+ ) is a Gelfand pair then so is (G n, K n ). Theorem.2. If Γ is non-abelian then (G Γ, K Γ ) fails to be a Gelfand pair. Thus for Γ non-abelian there is some integer 2 < N(Γ) Γ for which

GELFAND PAIRS ON WREATH PRODUCTS 3 (K n, G n ) is a Gelfand pair for all n < N(Γ) but (K n, G n ) fails to be a Gelfand pair for all n N(Γ). Examples.3. The authors of [AC2] used the GAP computer algebra system to verify that N(S 3 ) = 6, N(A 4 ) = 4, N(GL(2, F 3 )) = 3 and N(SL(3, F 2 )) = 3. We remar that we do not now whether or not N(Γ) can be arbitrarily large. Proofs for Theorems. and.2 are given below in Sections 2 and 4. Section 3 concerns decomposition of the spaces L(Γ n / n ) and L(G n /K n ) under the left actions of Γ n and G n. Section 5 concerns examples. We show that for primes p 3 the dihedral groups D p have N(D p ) = 6. As D 3 = S3 this is consistent with [AC2]. In [AM03] the reader will find a different family of Gelfand pairs involving wreath products with dihedral groups. 2. Proof of Theorem. For f L(Γ n+ ) let f L(Γ n ) be defined as and consider the map This is an algebra map. That is, f (x,..., x n ) = γ Γ f(x,..., x n, γ) Ψ : L(Γ n+ ) L(Γ n ), Ψ(f) = f. Lemma 2.. (f g) = f g for f, g L(Γ n+ ). Proof. In fact (f g) (x,..., x n ) = γ Γ f g(x,..., x n, γ) = γ Γ = y,...,y n+ Γ y,...,y n y n+ = f(x y,..., x nyn, γyn+ )g(y,..., y n+ ) f(x y,..., x nyn, γyn+ )g(y,..., y n+ ) γ y,...,y n y n+ γ f(x y,..., x ny n, γ )g(y,..., y n+ ) = f (x y,..., x nyn )g(y,..., y n+ ) y,...,y n y n+ = f (x y,..., x nyn )g (y,..., y n ) y,...,y n = f g (x,..., x n ). Lemma 2.2. Ψ ( L(Γ n+ ) Sn+ ) L(Γ n ) Sn. Proof. Say f L(Γ n+ ) Sn+ and σ S n. Then f (x σ(),..., x σ(n) ) = γ Γ f(x σ(),..., x σ(n), γ) = γ Γ f(x,..., x n, γ) = f (x,..., x n ) since (x σ(),..., x σ(n), γ) is a permutation of (x,..., x n, γ).

4 C. BENSON AND G. RATCLIFF For each m the group Γ Γ acts on the set Γ m via (γ, γ 2 ) (x,..., x m ) = (γ x γ2,..., γ x n γ2 ) and on L(Γ m ) via (γ, γ 2 ) f(x) = f ( (γ, γ 2 ) x). Lemma 2.3. Ψ is (Γ Γ)-equivariant. That is, ( (γ, γ 2 ) f) = (γ, γ 2 ) f for f L(Γ n+ ), γ, γ 2 Γ. Proof. Indeed ( (γ, γ 2 ) f) (x,..., x n ) = f(γ x γ 2,..., γ x nγ 2, γ γγ 2) γ Γ = γ Γ f(γ x γ 2,..., γ x nγ 2, γ ) = f (γ x γ 2,..., γ x nγ 2 ) = (γ, γ 2 ) f (x,..., x n ). Corollary 2.4. Ψ ( L( n+ \Γ n+ / n+ ) L( n \Γ n / n ). Proof. As L( m \Γ m / m ) = L(Γ m ) Γ Γ for all m this follows from Lemma 2.3. Recalling that A m (Γ) = L( m \Γ m / m ) L(Γ m ) Sm (see Equation.2) Corollary 2.4 together with Lemma 2.2 give the following. Corollary 2.5. Ψ ( A n+ (Γ) ) A n (Γ). We wish to show that in fact Lemma 2.6. Ψ ( A n+ (Γ) ) = A n (Γ). That is, Ψ : A n+ (Γ) A n (Γ) is surective. Woring towards a proof for this we introduce, for each m, the proection map P m : L(Γ m ) Sm A m (Γ), P m (f) = (γ, γ 2 ) f. As Ψ is (Γ Γ)-equivariant (Lemma 2.3) the diagram Ψ γ,γ 2 Γ L(Γ n+ ) Sn+ L(Γ) Sn P n+ P n A n+ (Γ) Ψ A n (Γ) commutes. As P n is surective we see that to prove Lemma 2.6 it suffices to show that Ψ : L(Γ n+ ) Sn+ L(Γ n ) Sn is surective. For this we require Lemma 2.7 below. List the elements of Γ as Γ = {γ,..., γ r } say where r = Γ. For x Γ m we let [x] := S m x denote the S m -orbit though x. This contains a unique point in which any γ s appear first followed by any γ 2 s etcetera in order. So [x] =, 2,..., r := [ ] γ,..., γ }{{}, γ 2,..., γ 2,..., γ }{{} r,..., γ }{{} r 2 r

GELFAND PAIRS ON WREATH PRODUCTS 5 for some non-negative integers,..., r with + + r = m. The characteristic functions { if x, δ, 2,..., r (x) = 2,..., r 0 otherwise give a basis for L(Γ m ) Sm. Lemma 2.7. For integers,..., r 0 with + + r = n + we have δ =, 2,..., r δ,...,,..., r. r 0 Proof. For x = (x,..., x n ) Γ n we have δ, 2,..., r (x) = r δ, 2,..., r (x,..., x n, γ ). = Here δ, 2,..., r (x,..., x n, γ ) = if and only if > 0 and x is a permutation of ( ) γ,..., γ }{{},..., γ,..., γ,..., γ r,..., γ r. }{{}}{{} r Thus δ, 2,..., r (x,..., x r, γ ) = if and only if > 0 and δ,...,,..., r (x) =. This completes the proof. Proof of Lemma 2.6. As explained above it suffices to show that Ψ : L(Γ n+ ) Sn+ L(Γ n ) Sn is surective. For this we will verify that δ,..., r Ψ ( ) L(Γ n+ ) Sn+ for all,..., r 0 with + + r = n. We do this by reverse induction on {0,..., n}. First suppose that = n so that,..., r = n, 0,..., 0. As δ n+,0,...,0 = δ n,0,...,0 this shows that δ,..., r Ψ ( L(Γ n+ ) Sn+ ) when = n. Next suppose that 0 n and assume inductively that δ,..., r Ψ ( ) L(Γ n+ ) Sn+ for all,..., r 0 with + + r = n and = +. Lemma 2.7 shows that δ = δ +, 2,..., r, 2,..., r + δ +,...,,..., r. 2 r 0 By inductive hypothesis all terms in the sum belong to Ψ ( ) L(Γ n+ ) Sn+ and hence so does δ, 2,..., r as desired. Proof of Theorem.. Suppose that (G n+, K n+ ) is a Gelfand pair. Equivalently the algebra A n+ (Γ) commutes under convolution. Given f, g A n (Γ) Lemma 2.6 ensure that there exist functions F, G A n+ (Γ) with F = f and G = g. Applying Lemma 2. now yields f g = F G = (F G) = (G F ) = G F = g f. Thus A n (Γ) is commutative and hence (G n, K n ) is a Gelfand pair.

6 C. BENSON AND G. RATCLIFF 3. Decomposition of L(Γ n / n ) and L(G n /K n ) 3.. Decomposition of L(Γ n / n ). One checs easily that the map Γ n / n Γ n, (x,..., x n ) n (x x n,..., x n x n ) is a well-defined biection. Using this to identify Γ n / n with Γ n the left quasiregular representation of Γ n on L(Γ n / n ) is realized on L(Γ n ) as ρ n : Γ n GL ( L(Γ n ) ) where (3.) ρ n (γ,..., γ n, γ )f(y,..., y n ) = f(γ y γ,..., γ n y n γ ). This is the restriction of the left-right regular representation of Γ n Γ n to Γ n n upon identification of the diagonal subgroup n Γ n with Γ. The Peter-Weyl Theorem shows that L(Γ n ) decomposes under Γ n Γ n as L(Γ n ) π π. π Γ n Here Γ n denotes the set of irreducible representations of Γ n modulo equivalence, π is the dual (or contragredient) representation for π and π π is an exterior tensor product representation for the product group Γ n Γ n. The irreducible representations for the product group Γ n are themselves exterior tensor products π π n of irreducible representations π Γ. The restriction of π π n to n = Γ is the interior tensor product representation π π n. For π n Γ let m(π,..., π n π n ) denote the multiplicity of π n in π π n so that π π n m(π,..., π n π n )π n. π n b Γ Now (3.) yields the decomposition (3.2) L(Γ n / n ) m(π,..., π n π n )π π n π n π,...,π n π n for L(Γ n / n ) as a Γ n -module. As (Γ n, n ) is a Gelfand pair if and only if L(Γ n / n ) is multiplicity free this proves the following. Proposition 3.. (Γ n, n ) is a Gelfand pair if and only if the interior tensor product representation π π n is multiplicity free for all irreducible representations π,..., π n Γ. In particular, taing n = 2 we recover the well-nown fact that (Γ Γ, 2 ) is a Gelfand pair. 3.2. Decomposition of L(G n /K n ). The irreducible representations for the wreath product G n = Γ n S n are constructed via the Macey machine as follows. Let π be an irreducible representation of Γ n. We have say π = π π n, the exterior tensor product of irreducible representations (π, V ) Γ. The stabilizer of π in S n, namely S π = {σ S n : π σ() = π for =,..., n}, acts on V V n via the intertwining representation ω : S π GL(V V n ), ω(σ)(v v n ) = v σ () v σ (n).

GELFAND PAIRS ON WREATH PRODUCTS 7 Given any ρ Ŝπ the induced representation ( R π,ρ = ind ) Gn Γ n S π (π ω) ρ is irreducible and every irreducible representation of G n is of this form. Example 3.2. Suppose we have n = 3, π = π π π 2. The stabilizer of π in S 3 is S π = S2 S. Letting ρ denote the trivial representation of S π the induced representation R π,ρ is R π,ρ = (π π π 2 ) (π π 2 π ) (π 2 π π ) as a representation of Γ 3 with S 3 permuting the factors. Recall that (G n = Γ n S n, K n = n S n ) is a Gelfand pair if and only if the space R Kn of K n -fixed vectors in R has dim(r Kn ) for every irreducible representation R Ĝn. Abusing terminology we call dim(r Kn ) the number of K n -fixed vectors in R. Letting K π = n S π for given π Γ n one has R π,ρ Kn = ind Kn K π ( (π ω) ρ ). An application of Frobenius reciprocity now yields the following. Lemma 3.3. The number of K n -fixed vectors in R π,ρ is equal to the number of K π -fixed vectors in (π ω) ρ. Thus in order for R π,ρ to be K n -spherical there must be K π -fixed vectors in (π ω) ρ and hence n -fixed vectors in π = π π n. Now (3.2) yields the following necessary condition for R π,ρ to be spherical. Lemma 3.4. If R π,ρ is K n -spherical then π n occurs in the (internal) tensor product π... π n. We will mae use of this criterion in connection with examples in Section 5. 4. Proof of Theorem.2 Among the irreducible representations for G n = Γ n S n discussed above are the following. Given an irreducible representation (π, V ) of Γ one obtains an irreducible representation π of G n in the n th tensor power W = n V via π(x, σ)(v v 2 v n ) = π(x )v σ () π(x 2 )v σ (2) π(x n )v σ (n) on decomposable tensors. Observe that the space of π(s n )-invariant vectors in W is W Sn = S n (V ), the n th symmetric power of V. The action of the diagonal subgroup n on W via π preserves W Sn as n and S n commute in G n. Moreover the representation ( π n, W Sn = S n (V )) of n coincides with (S n (π), S n (V )), the n th symmetric power of the representation (π, V ), under the obvious isomorphism n = Γ. It follows that the space W K n of (K n = n S n )-invariant vectors in W is precisely W Kn = S n (V ) Γ, the space of Γ-invariant vectors in the n th symmetric power of (π, V ). Here eπ = Rπ b b π, Sn in the notation from the previous section.

8 C. BENSON AND G. RATCLIFF Proof of Theorem.2. Let Γ be a finite non-abelian group of order r = Γ and (π, V ) an irreducible representation of Γ with d = dim(v ) >. Such a representation exists as Γ is non-abelian. A fundamental result in Invariant Theory asserts that the algebra S(V ) Γ of Γ-invariants in S(V ) contains d algebraically independent homogeneous elements. In fact Proposition 3.4 in [Sta79] shows that one can construct d such elements with degrees all dividing r. Suitable powers of these yield d linearly independent vectors in S r (V ) Γ. Letting ( π, W ) be the irreducible representation of G r = Γ r S r described above we now have that the space W Kr of (K r = r S r )-fixed vectors in W has dimension at least d. Thus (G r, K r ) fails to be a Gelfand pair as was to be shown. Remar 4.. The result that S(V ) Γ contains d algebraically independent homogeneous elements does not require that (π, V ) be irreducible. Most of the literature on Invariant Theory, including [Sta79], concerns invariants in polynomial rings C[V ] rather than symmetric algebras S(V ). To pass to this context one can replace (π, V ) above by its dual representation (π, V ) and note that S(V ) is canonically isomorphic to C[V ] as a Γ-module. 5. Wreath products with dihedral groups In this section we tae Γ = D p, where p is any odd prime. We show that (Γ n S n, n S n ) is a Gelfand pair for n 5 and not a Gelfand pair for n 6. The strategy is to first identify representations of Γ n which occur in L(Γ n / n ) by finding the tensor product representations which have n -fixed vectors. 5.. The case n = 3. We begin by reviewing the representation theory of D p. The conugacy classes are the identity element {I}, pairs of rotations {R, R }, and the set S of all reflections. There are two one-dimensional irreducible representations, the trivial representation θ and the determinant θ 2. In addition, there are m = (p )/2 two-dimensional representations π with characters χ. In each of these representations, any rotation has eigenvalues λ and λ, where λ is a pth root of unity. The character table is: () (2) (2)... (m) I R R 2... S θ... θ 2... χ 2 µ µ 2... 0 The second row of this table lists the conugacy classes, while the row above indicates the number of elements in each conugacy class. In the last row, π is a two-dimension representation, with µ = λ + λ for =,..., m = (p )/2. Since all of the characters are real-valued, by Lemma 3.4 we consider representations π i π π, where π occurs in π i π. Equivalently, we want π i π π to contain a 3 -fixed vector. Let us consider some examples. The representations π π and π π 2 have characters I R R 2... S χ 2 4 µ 2 µ 2 2... 0 χ χ 2 4 µ µ 2 µ 2 µ 4... 0

GELFAND PAIRS ON WREATH PRODUCTS 9 A calculation shows that µ i µ = µ i+ + µ i, where we understand that the subscripts are taen mod p, that µ = µ, and that µ 0 = 2. Thus in our examples, we get the characters I R R 2... S χ 2 4 µ 2 + 2 µ 4 + 2... 0 χ χ 2 4 µ + µ 3 µ 2 + µ 6... 0 and hence π π π 2 θ θ 2 and π π 2 π π 3. This tells us that the triples π π π 2, π π θ, π π θ 2 and π π 2 π 3 are all 3 -spherical representations of Γ 3. In each case, there is a single 3 -fixed vector, and hence each representation occurs in L(Γ 3 / 3 ) with multiplicity one. The spherical representations of Γ 3 are: π i π π i+ for i, π i π i π 2i π i π i θ π i π i θ 2 θ θ θ θ θ 2 θ 2 and all permutations of the tensor product factors. Each of these representations has a single 3 -fixed vector, hence it occurs in L(Γ 3 / 3 ) with multiplicity one. Note that for small values of p, not all cases may occur. Now we have: Proposition 5.. For Γ = D p, (Γ 3, 3 ) and (Γ 3 S 3, 3 S 3 ) are Gelfand pairs. 5.2. The case n = 4. We use identities of the following type to find the spherical representations: χ i χ χ = χ i±± ; χ 3 = χ (χ 2 + θ + θ 2 ) = χ + χ 3 + 2χ = 3χ + χ 3 ; χ 2 χ = (χ 2 + θ + θ 2 )χ = χ 2 + χ 2+ + 2χ. Representations of Γ 4 which occur in L(Γ 4 / 4 ): π i π π π i±± for i,, distinct. π (4) with multiplicity three. (Here π () denotes π π.) π (3) π 3, π (2) π (2) π π 2± with multiplicity two. θ i π 2. θ i θ for all choices of i,,. θ i π π π ± for. θ (4) i. θ (2) θ (2) 2. Since there are two cases of multiplicity, (Γ 4, 4 ) is not a Gelfand pair for Γ = D p.

0 C. BENSON AND G. RATCLIFF 5.3. Characters for Γ 4 S 4. Let π = π i π π π l be a representation of Γ 4, S π the stabilizer of π in S 4, and ω the intertwining representation of S π. Let χ π be the character for π ω. The technique for computing χ π is illustrated by the following example: Suppose that π = π (4), so that S π = S 4. Then χ π (δ, (234)) = χ (δ 4 ), χ π (δ, (23)) = χ π (δ 3 )χ π (δ), χ π (δ, (2)) = χ π (δ 2 )χ π (δ) 2, χ π (δ, (2)(34)) = χ π (δ 2 )χ π (δ 2 ). where we identify δ Γ with an element of 4. The two cases of multiplicity are π (4) and π (2), with stabilizers S 4 and S 2 S 2 respectively. Since π is determined by an arbitrary p th root of unity, we can tae = in both cases. We need to find the spherical representations of the form R π,ρ for ρ Ŝπ. That is, in view of Lemma 3.3, we see representations which contain (K π = 4 S π )- fixed vectors. The number of such vectors is: ( ) (5.) χ π (δ, σ)χ ρ (σ) = χ π (δ, σ) χ ρ (σ) 4 S π S π 4 δ,σ For each fixed σ S π, define the function σ m π,σ (δ) = χ π (δ, σ). This is a class function on = 4, which can be expressed as a linear combination of irreducible characters. So the sum M π (σ) = χ π (δ, σ) = m π,σ, δ is the coefficient of the trivial character θ in m π,σ. Moreover, the function M π is a class function on S π, so the sum (5.) is the coefficient of χ ρ = χ ρ in M π. For π = π (4), a straightforward calculation shows that: χ π (δ, e) = χ (δ) 4 = (χ 4 + 4χ 2 + 3θ + 3θ 2 )(δ), χ π (δ, (2)) = χ (δ 2 )χ (δ) 2 = (χ 4 + 2χ 2 + θ + θ 2 )(δ) χ π (δ, (23)) = χ (δ 3 )χ (δ) = (χ 4 + χ 2 )(δ) χ π (δ, (234)) = χ (δ 4 ) = (χ 2 + θ θ 2 )(δ) χ π (δ, (2)(34)) = χ (δ 2 )χ (δ 2 ) = (χ 4 + 3θ 2θ 2 )(δ) We tae the coefficient of θ to obtain: M π (e) = 3, M π ((2)) =, M π ((23)) = 0, M π ((234)) =, M π ((2)(34)) = 3. Consulting the character table of S 4, we find that M π is the sum of the trivial character and a two-dimensional character, with corresponding representations ρ and ρ 2. This tells us that (π (4) ω) ρ and (π (4) ω) ρ 2 are K π -spherical representations, each occurring with multiplicity one. Together, these spaces account for the occurrence of π (4) For π = π (2) with multiplicity three. and S π = S2 S 2, we obtain M π (e) = 2, M π ((2)) = 0, M π ((34)) = 0, M π ((2)(34)) = 2. δ

GELFAND PAIRS ON WREATH PRODUCTS Thus M π is the sum of the trivial character ρ and the sign character ρ s. This tells us that (π (2) ω) ρ and (π (2) ω) ρ s are spherical representations, each occurring with multiplicity one. Together, these spaces account for the occurrence of π (2) with multiplicity two. Thus we see that (Γ 4 S 4, S 4 ) is a Gelfand pair. 5.4. The case n = 5. Using similar techniques to those above, we can find the spherical representations of Γ 5. We list ust those with multiplicity: π (4) θ i with multiplicity 3. π (2) π (4) π (3) π (2) π (2) θ i with multiplicity 2. π 2 with multiplicity 4. π π ± with multiplicity 3. π 2 with multiplicity 2. π π l π ±l with multiplicity 2. Since χ 4 θ i = χ 4 and χ2 χ2 θ i = χ 2 χ2, the first two cases are handled in the previous section. For the other cases, we tae =. For π = π (4) π 2 and S π = S4 S, we obtain M π (e) = 4, M π ((2)) = 2, M π ((23)) =, M π ((2)(34)) = 0, M π ((234)) = 0. Thus M π is the character of the standard 4-dimensional representation of S 4, which is the sum of the trivial representation ρ and a 3-dimensional irreducible representation ρ 3. This tells us that (π (4) π 2 ω) ρ and (π (4) π 2 ω) ρ 3 are spherical representations, each occurring with multiplicity one. Together, these spaces account for the occurrence of π (4) π 2 with multiplicity four. For π = π (3) π π + and S π = S3, we obtain M π (e) = 3, M π ((2)) =, M π ((23)) = 0. Thus M π is the character of the standard 3-dimensional representation of S 3, which is the sum of the trivial representation ρ and a 2-dimensional irreducible representation ρ 2. This tells us that (π (3) π π + ω) ρ and (π (4) π π + ω) ρ 2 are spherical representations, each occurring with multiplicity one. Together, these spaces account for the occurrence of π (4) π π + with multiplicity three. For π = π (2) π 2 and S π = S2 S 2, we obtain M π (e) = 2, M π ((2)) = 2, M π ((34)) = 0, M π ((2)(34)) = 0. Thus M π is the sum of two -dimensional irreducible representations of S 2 S 2, one of which is the trivial representation. For π = π (2) π π l π ±l and S π = S2, we obtain M π (e) = 2, M π ((2)) = 0. Thus M π is the sum of the two -dimensional irreducible representations of S 2. We conclude that (Γ 5 S 5, 5 S 5 ) is a Gelfand pair.

2 C. BENSON AND G. RATCLIFF 5.5. The Case n = 6. In this case, (Dp 6 S 6, 6 S 6 ) fails to be a Gelfand pair. This is consistent with the GAP-generated result for p = 3 in [AC2]. Let π = π (4) 2, with S π = S 4 S 2. Then we obtain M π (e) = 7, M π ((2)) = 3, M π ((23)) =, M π ((234)) =, M π ((2)(34)) = 3, M π ((56)) =, M π ((2)(56)) =, M π ((23)(56)) =, M π ((234)(56)) = 3, M π ((2)(34)(56)) = 5. ( ) One can see that M π, ρ = 2, and hence that R π,ρ = Ind G6 Γ 6 S π (π ω) ρ has multiplicity 2 in L(G 6 /K 6 ). For p = 3, we have π = π 2, and the result holds for π = π (6). References [AC2] Kürşat Aer and Mahir Bilen Can. From paring functions to Gelfand pairs. Proc. Amer. Math. Soc., 40(4):3 24, 202. [AM03] Hiroi Aazawa and Hiroshi Mizuawa. Orthogonal polynomials arising from the wreath products of a dihedral group with a symmetric group. J. Combin. Theory Ser. A, 04(2):37 380, 2003. [CSST06] Tullio Ceccherini-Silberstein, Fabio Scarabotti, and Filippo Tolli. Trees, wreath products and finite Gelfand pairs. Adv. Math., 206(2):503 537, 2006. [CSST08] Tullio Ceccherini-Silberstein, Fabio Scarabotti, and Filippo Tolli. Harmonic analysis on finite groups, volume 08 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2008. Representation theory, Gelfand pairs and Marov chains. [CSST4] Tullio Ceccherini-Silberstein, Fabio Scarabotti, and Filippo Tolli. Representation theory and harmonic analysis of wreath products of finite groups, volume 40 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 204. [Mac95] I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New Yor, second edition, 995. [Sta79] Richard P. Stanley. Invariants of finite groups and their applications to combinatorics. Bull. Amer. Math. Soc. (N.S.), (3):475 5, 979. [Ter99] Audrey Terras. Fourier analysis on finite groups and applications, volume 43 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 999. Dept of Mathematics, East Carolina University, Greenville, NC 27858 E-mail address: bensonf@ecu.edu, ratcliffg@ecu.edu