1999/03/27 by Manabu Yamaguchi, Yamaguchi, Manabu
Mathematics · #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.math/9903159
AMS-TeX, 22 pages, submitted to Advanced Studies in Pure Math
arxiv created 1999/03/30 · arxiv updated 2009/11/30
We establish a duality relation between one of the twisted group algebras of the hyperoctahedral groupf Hk and a Lie superalgebra q(n0) ⊕ q(n1) for any integers k and n0, n1, where q(n0) and q(n1) denote the ``queer'' Liesuperalgebras. Note that this twisted group algebra \B'k belongs to a different cocycle from the one \Bk used by A. N. Sergeev in [8] and by the present author in [11]. We will use the supertensor product \Ck ⊗ \B'k of the 2k-dimensional Clifford algebra \Ck and \B'k, as an intermediary for establishing our duality. We show that the algebra \Ck ⊗ B'k and q(n0) ⊕ q(n1) act on the k-fold tensor product W=V⊗ k of the natural representation V of q(n0+n1) ``as mutual centralizers of each other'' (Theorem 4.1). Moreover, we show that \B'k and q(n0) ⊕ q(n1) act on a subspace W' of W ``as mutual centralizers of each other'' (Theorem 4.2). This duality relation gives a formula for character values of simple B'k-modules. This formula is di fferent from a formula (Theorem D) obtained by J. R. Stembridge (cf. [10, Lem 7.5]).