Irreducible projective characters of wreath products

The irreducible character values of the spin wreath products of the symmetric group and an arbitrary finite group are completely determined.


Introduction
The symmetric group has been one of the core materials in mathematics and theoretical physics since Frobenius developed representation theory of finite groups. Using the duality between S n and GL m Schur found the complete set of invariants for the general linear group and revolutionized the theory of symmetric functions using the Schur functions in his dissertation. In the thirties Specht [13] generalized Schur and Frobenius' theory to the wreath products of the symmetric group and any finite group. More recently Macdonald reformulated Specht's theory in the classic monograph [6].
The double covering groups S n of the symmetric group have many similar and interesting properties as shown by Schur in [11]. In that seminal paper Schur generalized Frobenius's theory and introduced the famous Schur Qfunctions. Schur's character theory of S n consists of two parts. The first part of the character values on conjugacy classes associated to partitions with odd integer parts are exactly given by an analogous Frobenius formula in terms of the Schur Q-functions; the second part of character values on strict partitions was solved with the help of twisted tensor products of Clifford algebras. In the same direction, Morris [7] formulated an iterative rule for computing the spin character values of the symmetric group, and Nazarov [9] constructed all irreducible representations of the spin group. Józefiak [5] also computed the projective character values for a related double covering group of the hyperoctahedral group using similar techniques.
In [1] the second author and collaborators determined all irreducible character values on even conjugacy classes of the spin wreath products Γ n by vertex operator calculus in the context of the spin McKay correspondence. The character values generalize the first part of Schur's theory on S n . The key was to show that the character values at conjugacy classes of even colored partitions (with odd integer parts) are given by matrix coefficients of certain products of twisted vertex operators. However, the other character values on odd strict partition-valued functions could not be explained in the context of the McKay correspondence. To the authors' knowledge, other available methods such as the Hopf algebraic approach [2] seem not helpful either.
The knowledge of the remaining character values for Γ n would be an analog of the second part in Schur's pioneering work [11]. In the case of Γ being abelian, we solved the problem in [3] by using the Mackey-Wigner method of little groups (cf. [12]) to construct all irreducible spin representations of the wreath products (see also [8] for some cases). However the method of little groups does not work in the most general case for arbitrary finite group Γ. It seems that a new method is needed for determination of the irreducible characters.
The purpose of this paper is to complete the character theory of the spin wreath products Γ n and compute the missing part of the characters table of Γ n for any finite group Γ. We will construct all irreducible characters and in particular provide explicit formulas for character values on the conjugacy classes of the second type. It turns out that spin character values in this part can be non-zero for more than two conjugacy classes in contrast of Schur's case, nevertheless they are still sparsely zero. We note that the exhaustion method is used to pick up all non-zero character values, which bears some similarity to Schur's original method.
2. Projective representations of Γ n 2.1. Spin wreath products. According to Schur, there are two nonisomorphic double covering groups of the symmetric group S n when n ≥ 4 and n = 6. But their representations are in complete one-to-one correspondence. We fix one of them, and let the spin symmetric group S n be the finite group generated by z and t i , (i = 1, · · · , n − 1) with the relations: Let θ n be the homomorphism from S n to S n sending t i to the transposition (i, i + 1) and z to 1. This says that S n is a central extension of S n by the cyclic group Z 2 .
The spin group S n has a cycle presentationà la Conway [14]. For i < j, the transposition [ij] is deinfed as [ij] = t j−1 · · · t i+1 t i t i+1 · · · t j−1 . Here if j = i + 1, we take t i = [i, i + 1]. Then for i 1 < i 2 < · · · < i k ≤ n, we Finally for a permutation σ ∈ S k we define Given a permutation w ∈ S n . We can fix its cycle product as follows. First each cycle is written as a word lexicographically by rotating its content, then we rearrange the order of the cycles lexicographically to obtain a unique presentation w = l i=1 (a i1 · · · a iλ i ), where λ i are the lengths of the cycles. We then define the element t w = l i=1 [a i1 · · · a iλ i ] ∈ S n . Note that θ −1 n (w) = z p t w . One also has that t w 1 t w t −1 for w, w 1 ∈ S n , where p = 0 or 1. Similarly for a partition ρ we denote t ρ = t w(ρ) , where w(ρ) = (1 · · · ρ 1 ) · · · (n − ρ l + 1 · · · n).
This agrees with the usual parity for S n when Γ is the trivial group. Therefore the group algebra C[ Γ n ] has a superalgebra structure, and C[Γ ≀ A n ] is the even subspace.

Conjugacy classes.
Let Γ * = {c i |i = 0, 1, . . . , r} be the set of conjugacy classes of Γ and denote by Γ * = {γ i |i = 0, 1, · · · , r} the set of irreducible characters of Γ. Let ζ c be the order of the centralizer of an element in the conjugacy class c ∈ Γ * , then the order of the conjugacy class c is |Γ|/ζ c . Here for a finite set X we denote by |X| its cardinality. In the following we follow Macdonald's notations [6].
A partition-valued function ρ = (ρ(c)) c∈Γ * defined on Γ * consists of |Γ * | partitions indexed by conjugacy classes c ∈ Γ * . The weight of ρ is defined by ||ρ|| = c∈Γ * |ρ(c)|, and the length is given by l(ρ) = c∈Γ * l(ρ(c)). It helps to visualize ρ as a colored partition in which each sub-partition ρ(c) is colored by c. Let P(Γ * ) be the set of partition-valued functions indexed by Γ * . It is well-known that the conjugacy classes of Γ n are parameterized by P(Γ * ). For an element (g, σ) ∈ Γ n , the permutation σ gives rise to a cycle partition λ = (λ 1 , λ 2 , . . .). For each part λ i = k, which corresponds to the cycle (i 1 i 2 · · · i k ), we associate the cycle-product g i k g i k−1 · · · g i 1 ∈ Γ. If the cycle-product belongs to the conjugacy class c, then we color this part λ i by c which turns λ into a colored partition. In this way we get the parametrization of conjugacy classes of Γ n by P(Γ * ). For ρ = (ρ(c)) c∈Γ * ∈ P n (Γ * ), let C ρ be the corresponding conjugacy class in Γ n .

Split conjugacy classes. An element
A conjugacy class of Γ n is called split if its elements are split. It is known that the conjugacy class C ρ of Γ n splits if and only if the preimage θ −1 n (C ρ ) =: D ρ splits into two conjugacy classes in Γ n .
For a partition λ = (1 m 1 2 m 2 3 m 3 · · · ) of n, we denote by z λ = i≥1 i m i m i ! the order of the centralizer of the permutation with cycle type λ in S n . For each partition-valued function ρ = (ρ(c)) c∈Γ * , we define which is the order of the centralizer of an element of conjugacy type ρ = (ρ(c)) c∈Γ * in Γ n . The order of the centralizer of an element of conjugacy type ρ in Γ n is given by For each split conjugacy class C ρ in Γ n , we define the conjugacy class D + ρ in Γ n to be the conjugacy class containing the element (g, t ρ ) and define As usual a representation π of Γ n is called spin if π(z) = −id, then its character is a projective character of Γ n . It is clear that the character of a spin representation of Γ n are determined by its values on split conjugacy classes. By a standard result [10,1] the conjugacy class of Γ n is split in Γ n if and only if either ρ ∈ OP n (Γ * ) or ρ ∈ SP 1 n (Γ * ). In particular the split conjugacy classes of S n (i.e. when Γ is the trivial group) are parameterized either by partitions with odd integers or by odd strict partitions. By Euler's theorem the number of strict partitions is equal to the number of odd partitions, therefore the total number of such conjugacy classes of Γ n are given by |SP 0 3. The irreducible spin character table of Γ n 3.1. Schur's theory of S n . Like the symmetric group S n , nontrivial projective (spin) characters of S n are parameterized by strict partitions λ of n. One can classify spin characters into the so-called double spin and associate spin characters. The associated character χ ′ of a spin character is defined to be χ ′ = sgn · χ, where sgn is the sign character. If χ ′ = χ, then we say χ is a double spin character (or self-associated). For ν ∈ SP n and d(ν) = n − l(ν) even, there corresponds a unique irreducible (double) spin character ∆ ν ; for d(ν) odd, there corresponds a pair of irreducible (associate) spin characters ∆ ± ν for S n . Schur [11] showed that there was an analogous Frobenius formula for the spin character values at the even conjugacy classes indexed by partitions with odd integers. For ν ∈ SP the Schur Q-function Q ν is defined by where n ≥ l = l(ν) and S n−l acts on x l+1 , · · · , x n . It is known that Q ν is a polynomial in power sum symmetric functions p 1 , p 3 , p 5 , · · · . As usual we will denote p α = p α 1 p α 2 · · · for a partition α. Schur showed that nontrivial values of the spin character ∆ ν at even conjugacy classes are given by where [a] denotes the largest integer ≤ a andd(ν) is equal to 0 (resp. 1) if d(ν) is even (resp. odd). Schur further proved the following results.

Decomposition of colored partitions.
When Γ is a finite abelian group, we used the Mackey-Wigner method of little groups to decompose the action of S n on the characters of Γ n . It turns out that the invariant subgroup of each S n -orbit is a Young subgroup of S n and vice versa. Then we can construct all spin irreducible representations indexed by strict partition-valued functions by induction, and show that the character values are sparsely zero and the non-zero values are given according to how the partitions are supported on various conjugacy classes (see [3]). This method is no longer available when Γ is an arbitrary finite group. Next we use a different method to compute spin character values on odd strict partition-valued functions for a general finite group Γ.
3.3. Spin supermodules vs. spin modules. A spin Γ n -module V becomes a spin supermodule when ch V (x) = 0 for all odd elements x. According to [5] there are two basic types of simple supermodules: type M or Q, corresponding to our double spin and a pair of associated spin modules when forgetting the Z 2 -gradation. Moreover all double spin and associate spin modules are realized in this way.
Proof. This is true in a more general context. Each irreducible Γ nsupermodule of type M (resp. Q) is an irreducible double spin (resp. a pair of associated spin) Γ n -module(s). Suppose that the underlying Γ n -module of our irreducible Γ n -supermodule V = Ind Γn Γ λ W λ decomposes into a direct sum of irreducible Γ n -modules: where V i are irreducible double spin modules, and W j and W ′ j are irreducible associate spin modules. It follows from general theory [4] of double spin and associate spin modules that, as an Γ ≀ A n -module, On the other hand we know that V, V Γn = 1 or 2 according to the spin supermodule V being of type M or Q by vertex operator calculus [1]. This means that V, V Γ≀ An = 2, so we must have that either m = 1 or q = 1.
Proof. Write m = |J λ | then we have (Note: here m is odd) As Z ρ = γ∈Γ * Z ργ and Z ργ = 2Z ργ for γ ∈ J λ , so Eq. (3.6) becomes (3.7) Thus it follows from [1] and Lemma 3.2 that Eqs. (3.6) and (3.7) imply that The following theorem gives the remaining part of the spin character table for Γ n .