Remarks on the Schur-Howe-Sergeev Duality

TL;DR

This paper introduces a new Howe duality involving queer Lie superalgebras, linking representation theory with combinatorial identities for Schur Q-functions and connecting it to the Schur-Sergeev duality with hyperoctahedral groups.

Contribution

It establishes a novel Howe duality between two queer Lie superalgebras and connects it to the Schur-Sergeev duality, providing new insights into representation theory and combinatorial identities.

Findings

New Howe duality between q(m) and q(n) superalgebras.

Representation-theoretic interpretation of Schur Q-functions.

Equivalence with Schur-Sergeev duality involving hyperoctahedral groups.

Abstract

We establish a new Howe duality between a pair of two queer Lie superalgebras (q(m),q(n)). This gives a representation theoretic interpretation of a well-known combinatorial identity for Schur Q-functions. We further establish the equivalence between this new Howe duality and the Schur-Sergeev duality between q(n) and a central extension $\Hy_k$ of the hyperoctahedral group H_k. We show that the zero-weight space of a q(n)-module with highest weight $\lambda$ given by a strict partition of n is an irreducible module over the finite group $\Hy_n$ parameterized by $\lambda$. We also discuss some consequences of this Howe duality.