Gelfand Pairs Involving the Wreath Product of Finite Abelian Groups With

Home , Gelfand pair, Induced representation

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n In [1], Aker and Can prove that if Γ is a ﬁnite abelian group then (Γ ⋉ Sn, Sn) is a Gelfand pair where Γn = Γ × ... × Γ (n copies), Γn is the quotient of Γn fby its diagonal subgroup n n {(g,g,...,g) | g ∈ Γ} and Γ ⋉ Sn is the semidirectf product of Γ with Sn. In the particular case when Γ = Zr, the authorsf give explicit descriptions of the irreduciblef constituents of the fn Γ ⋉Sn multiplicity free representation IndSn (1).

In [10], Stein presents a complete list of irreducible representations of the wreath product G ≀ Sn where G is a ﬁnite group. In particular, he gives a branching rule for the group G ≀ Sn similar to that of the symmetric group. Using this rule we will show the following main result of this paper

Theorem 1.1. Let G be a finite group then the pair (G ≀ Sn,G ≀ Sn−1) is a Gelfand pair if and only if G is abelian. We will only use the necessary definitions and facts from the representation theory of finite groups to prove this theorem. We eschew unuseful details and give references to the needed results instead of giving proofs. By doing so, we hope to present this result in a clear and short way. The paper is organised as follows. In Section 2, we review some useful tools from the theory of representation of finite groups and we define Gelfand pairs. In Section 3, we define multipartitions and we present the irreducible representations of the wreath product of a finite group G by Sn. In Section 4, we prove Theorem 1.1 the main result of this paper.

2. REPRESENTATION THEORY OF FINITE GROUPS AND GELFAND PAIRS In this paper we only consider representations over the complex numbers field C. We review in this section some definitions and well known results in the theory of representations of finite groups. We only present the necessery results that are useful to prove our main result. The reader is invited to check the book [9] of Sagan for complete proofs.

2.1. Induced representations. Let G be a ﬁnite group. A representation of G is a pair (V, ρ) where V is a ﬁnite dimensional vector space over C and ρ : G −→ GL(V ) is a group homomorphism. Here GL(V ) stends for the group of all invertible linear maps from V to V. The dimension of a representation is the dimension of its vector space. For example, the trivial representation of G is the one-dimensional representation (C, 1) where 1(g)(c)= c for any g ∈ G and c ∈ C. Let K be a subgroup of G, G/K = {g1,...,gr} be a set of representatives of the left cosets of K in G and (V, ρ) be a representation of K. Using (V, ρ), we can make a representation G G (IndK V, IndK ρ) of G, called the induced representation of (V, ρ) to G where r IndG V = g V K M i i=1 with giV being an isomorphic copy of V whose elements are written as giv with v ∈ V. For each g ∈ G and each gi there is an hi ∈ K and j(i) ∈ {1,...,r} such that ggi = gj(i)hi and GELFANDPAIRSINVOLVINGFINITEABELIANGROUPS 3

G IndK ρ(g) is defined by: r r IndG ρ(g) g v = g ρ(h )(v ). K X i i X j(i) i i i=1 i=1 For example, in the particular case of the trivial representation of K, the induced represen- C G C tation can be identified with the representation ( [G/K], 1K) of G, where [G/K] is the r- dimensional vector space spanned by the left cosets giK and G 1K(g)(kiK)=(gki)K for any g ∈ G and any representative gi of G/K. 2.2. Facts from representation theory of finite groups. A vector subspace W of V is said to be a subrepresentation of (V, ρ) if ρ(g)(w) ∈ W for any w ∈ W. The representation (V, ρ) is said to be reducible if it has a subrepresentation other than {0} and V. Otherwise it is said to be irreducible. If (V1, ρ1) and (V2, ρ2) are two representations of G then (V1 ⊕ V2, ρ1 ⊕ ρ2), where V1 ⊕ V2 is the direct sum of V1 and V2, is a representation of G with:

(ρ1 ⊕ ρ2)(g)(v1 + v2) := ρ1(g)(v1)+ ρ2(g)(v2),

for any g ∈ G, v1 ∈ V1 and v2 ∈ V2. A representation (V, ρ) is equivalent to an action of G on V, that is gv = ρ(g)(v) for any g ∈ G and v ∈ V. To simplify notations, it will be convenient for us to omit the homomorphism and say that V is a representation of G. If V and W are two representations of G then θ : V → W is said to be a representations homomorphism (or an intertwiner) if θ is an homomorphism of vector spaces that satisﬁes θ(gv) = gθ(v) for any g ∈ G and any v ∈ V. Two representations are equivalent if there exists a representations isomorphism between them. Oth- erwise, they are called inequivalent representations. We will present below some well known facts in the theory of representation of ﬁnite groups. For complete proofs, the reader may refer to the book [9] of Sagan.

The number of inequivalent irreducible representations of G is equal to the number of conjugacy classes of G, see [9, Proposition 1.10.1]. Let (V, ρ) and (W, σ) be two irreducible representations. Then the Schur lemma (see [9, Theorem 1.6.5]) states that any representation homomorphism θ : V −→ W is either the zero map (if ρ 6∼ σ) or a vector space isomorphism (if ρ ∼ σ). As a consequence of the Schur lemma, if (V, ρ) and (W, σ) are two irreducible representations over the same space, then any representation homomorphism θ : V −→ V is either the zero map (if ρ 6∼ σ) or θ = c IdV for a certain nonzero c ∈ C, where IdV is the identity map on V (if ρ ∼ σ). Let {V 1,...,V l} be a complete list of irreducible pairwise inequivalent representations of G then by [9, Proposition 1.10.1] l |G| = (dim V i)2. X i=1 In addition, by Maschke’s theorem (see [9, Theorem 1.5.3]) any ﬁnite dimensional representation V of G can be decomposed as the direct sum of irreducible representations of G l V ≃ m V i, M i i=1 4 O. TOUT

i i i i where mi ∈ N and miV := V ⊕ V ⊕···⊕ V (mi copies) for any 1 ≤ i ≤ l. We say that the representation V of G is multiplicity free if mi ≤ 1 for any 1 ≤ i ≤ l. 2.3. Gelfand pairs. Let (G,K) be a pair where G is a ﬁnite group and K is a subgroup of G. Consider C(G,K), the algebra of invariant complex valued functions on the double cosets of K in G, that is C(G,K) := {f : G −→ C | f(kxk′)= f(x) for any x ∈ G and any k,k′ ∈ K}. The multiplication in C(G,K) is deﬁned by the convolution product of functions : (fg)(x)= f(y)g(y−1x) for any f,g ∈ C(G,K). X y∈G The pair (G,K) issaidtobea Gelfand pair if its associated algebra C(G,K) is commutative. Equivalently, (G,K) is a Gelfand pair if the algebra C[K \ G/K] spanned by the double cosets of K in G is commutative. In [8, (1.1) page 389], it is shown that the pair (G,K) is a Gelfand pair if and only if the induced representation C[G/K] of the trivial representation of K to G is multiplicity free.

3. MULTIPARTITIONS AND THE IRREDUCIBLE REPRESENTATIONS OF G ≀ Sn

In this section we present a complete list of inequivalent irreducible representations of G ≀ Sn for a ﬁnite group G. As in the previous section, we don’t give the proofs of the presented results. The reader is invited to read the paper [10] of Stein for more details. 3.1. Partitions. A partition λ is a weakly decreasing list of positive integers (λ1,...,λl) where λ1 ≥ λ2 ≥ ... ≥ λl ≥ 1. The λi are called the parts of λ; the size of λ, denoted by |λ|, is the sum of all of its parts. If |λ| = n, we say that λ is a partition of n. We will also use the m1(λ) m2(λ) m3(λ) exponential notation λ = (1 , 2 , 3 ,...), where mi(λ) is the number of parts equal to i in the partition λ. In case there is no confusion, we will omit λ from mi(λ) to simplify our m1(λ) m2(λ) m3(λ) mn(λ) n notation. If λ = (1 , 2 , 3 ,...,n ) is a partition of n then i=1 imi(λ) = n. mi(λ) P We will dismiss i from λ when mi(λ)=0. If λ =(λ1,...,λl) is a partition of n, there are several natural ways to extend it to a partition of n +1. This can be done by adding 1 to λ1, adding a part λl+1 equal to one, adding 1 to the parts λi that satisfy λi−1 > λi ≥ λi+1 for 2 ≤ i ≤ l − 1 or by adding 1 to the parts λi+1 that satisfy λi−1 = λi > λi+1 for 2 ≤ i ≤ l − 1. We will denote by λ+ the set of all extended partitions from λ. For example, (3, 3, 2, 2, 2, 1)+ = {(4, 3, 2, 2, 2, 1), (3, 3, 2, 2, 2, 1, 1), (3, 3, 3, 2, 2, 1), (3, 3, 2, 2, 2, 2)}.

3.2. Branching rule for Sn. It is well known that partitions of n are used to index all of the conjugacy classes and the irreducible representations of Sn. The irreducible representation of Sn that corresponds to the partition λ of n is called Specht module and usually denoted by Sλ, see [9, Theorem 2.4.6]. The branching rule shows that the inducing representation of an irreducible λ representation S of Sn−1 to Sn has a nice decomposition into irreducible representations of Sn. Explicitly, if λ is a partition of n − 1 then by [9, Theorem 2.8.3] we have the following formula:

IndSn Sλ = Sδ. Sn−1 M δ∈λ+ GELFANDPAIRSINVOLVINGFINITEABELIANGROUPS 5

(n−1) In particular, since the trivial representation of Sn−1 corresponds to the Specht module S by [9, Example 2.3.6], we have

Sn Sn (n−1) (n) (n−1,1) 1Sn−1 = IndSn−1 S = S ⊕ S and (Sn, Sn−1) is a Gelfand pair.

3.3. Irreducible representations and a branchig rule for G≀Sn. Let G be a ﬁnite group. The n wreath product G ≀ Sn is the group with underlying set G × Sn and product deﬁned as follows:

((σ1,...,σn); p).((ǫ1,...,ǫn); q)=((σ1ǫp−1(1),...,σnǫp−1(n)); pq), n for any ((σ1,...,σn); p), ((ǫ1,...,ǫn); q) ∈ G × Sn. Suppose now that {V 1,...,V l} is a complete list of irreducible pairwise inequivalent representations of G. We assume that V 1 is the trivial representation of G. An l-multipartition of n is a tuple Λ=(λ1,λ2,...,λl) where λi is a partition for each 1 ≤ i ≤ l and l |Λ| := |λ | = n. X i i=1

The irreducible representations of G ≀ Sn are indexed by l-multipartitions of n. We will denote Λ by S the irreducible representation of G≀Sn that corresponds to an l-multipartition Λ of n. The set {SΛ | Λ is an l-multipartition of n} forms a complete list of irreducible representations of G ≀ Sn, see for example [10]. In [10, Theorem 5.1], there is a branching rule for the group G ≀ Sn similar to that of the symmetric group Sn presented in the previous subsection. Explicitly, if Λ=(λ1,λ2,...,λl) is an l-multipartition of n − 1 then we have the following formula:

l ≀S IndG n SΛ = dim V i S(λ1,...,λi−1,δ,λi+1,...,λl). G≀Sn−1 M M i + =1 δ∈λi In particular,

G≀Sn G≀Sn (n−1),∅,∅,...,∅ (1) 1G≀Sn−1 = IndG≀Sn−1 S l = S (n),∅,∅,...,∅ ⊕ S (n−1,1),∅,∅,...,∅ dim V i.S (n−1),∅,...,∅,(1),∅,...∅,∅ , M i=2

th (n−1),∅,...,∅,(1),∅,...∅,∅ where (1) is in the i position of S .

4. PROOF OF THE MAIN RESULT

By Equation (1), the pair (G ≀ Sn,G ≀ Sn−1) is a Gelfand pair if and only if all the irreducible representations V i of G are one dimensional. In the following paragraph we will show that all the irreducible representations of a finite group G over the field of complex numbers C are one dimensional if and only if G is abelian. Thus, we prove Theorem 1.1. To start with, suppose that G is a finite abelian group and that (V, ρ) is an irreducible representation of G. For any g ∈ G, ρ(g): V → V is a G-homomorphism, that is ρ(g) ◦ ρ(h) = ρ(h) ◦ ρ(g) for any h ∈ G. This can be checked easily since ρ(g) ◦ ρ(h) = ρ(gh) = ρ(hg) = 6 O. TOUT

ρ(h) ◦ ρ(g). But (V, ρ) is irreducible, it follows then from Schur’s lemma that for any g ∈ G, ρ(g)= cg IdV , where cg ∈ C. Let v be a nonzero vector of V, the subspace ≺ v ≻ generated by v is G-invariant since for any av ∈≺ v ≻ with a ∈ C we have ρ(g)(av)= aρ(g)(v)= acgv ∈≺ v ≻ . Thus ≺ v ≻ is a subrepresentation of V which is irreducible. Therefore ≺ v ≻= V and dimC V =1. In the opposite direction, we have l |G| = (dim V i)2. X i=1 Thus if all the V i are one dimensional then the number of irreducible representations of G, which is l, is equal to |G|. Thus the number of conjugacy classes of G is |G|, that is the conjugacy class of any g ∈ G is {g}. This means that for any g, h ∈ G, hgh−1 = g or equivalently hg = gh and G is thus abelian.

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