A Duality of the Twisted Group Algebra of the Symmetric Group and a Lie Superalgebra

The “character values” of the irreducible projective representations of Sk, the symmetric group of degree k, were determined by I. Schur using Schur’sQ-functions, which are indexed by the distinct partitions of k, [10], in a way analogous to Frobenius’ formula for the character values of the ordinary irreducible representations of Sk [2]. Behind Frobenius’ formula exists a duality relation of Sk and the general linear group GL(n) (the Schur-Weyl duality). It is natural to expect the existence of an analogous duality relation between the twisted group algebra Ak (cf. (1.2)) of Sk and some algebra, behind Schur’s method. A. N. Sergeev showed that a twisted group algebra Bk (cf. (1.3)) of the hyperoctahedral group Hk and a Lie superalgebra q(n) (cf. §1, G) act on the k-th tensor product W = V ⊗k of the 2ndimensional natural representation V = C ⊕ C of q(n), as mutual commutants of each other [11] (in the sense of Z/2Z-graded algebras, see §1, E). This result motivated our work. In this paper, we establish a duality relation between Ak and q(n) on a subspace ofW , and give a representation-theoretic explanation of Schur’s identity (1.6) adapted to the context of Z/2Z-graded representations by T. Józefiak (Corollary 4.2). In §3, we construct an isomorphism Bk ∼= Ck . ⊗ Ak of Z/2Z-graded algebras (Theorem 3.2), where Ck is the 2-dimensional Clifford algebra and . ⊗ denotes the Z/2Z-graded tensor product (cf. §1, E). This isomorphism does imply an embedding Ak →֒ Bk, although Ak does not sit in Bk in an obvious manner (cf. §1, D). Then, we give a simple relation between the Z/2Z-graded irreducible representations of Bk and Ak (Proposition 3.5). Note that J. R. Stembridge constructed the non-graded simple modules of the underlying algebra |Bk| as submodules of non-graded tensor products of modules of three twisted group algebras of Hk [13],