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THE QUEER $q$-SCHUR SUPERALGEBRA

Part of: Linear algebraic groups and related topics

Published online by Cambridge University Press:  02 February 2018

JIE DU  and
JINKUI WAN
JIE DU
Affiliation:
School of Mathematics and Statistics, University of New South Wales, UNSW Sydney 2052, Australia email j.du@unsw.edu.au.
JINKUI WAN*
Affiliation:
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, PR China email wjk302@hotmail.com
*
✉wjk302@hotmail.com
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Abstract

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As a natural generalisation of $q$-Schur algebras associated with the Hecke algebra ${\mathcal{H}}_{r,R}$ (of the symmetric group), we introduce the queer $q$-Schur superalgebra associated with the Hecke–Clifford superalgebra ${\mathcal{H}}_{r,R}^{\mathsf{c}}$, which, by definition, is the endomorphism algebra of the induced ${\mathcal{H}}_{r,R}^{\mathsf{c}}$-module from certain $q$-permutation modules over ${\mathcal{H}}_{r,R}$. We will describe certain integral bases for these superalgebras in terms of matrices and will establish the base-change property for them. We will also identify the queer $q$-Schur superalgebras with the quantum queer Schur superalgebras investigated in the context of quantum queer supergroups and provide a constructible classification of their simple polynomial representations over a certain extension of the field $\mathbb{C}(\mathbf{v})$ of complex rational functions.

Keywords

Hecke–Clifford superalgebraq-Schur algebraqueer q-Schur superalgebraquantum queer supergroup

MSC classification

Primary: 20G43: Schur and $q$-Schur algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 105 , Issue 3 , December 2018 , pp. 316 - 346
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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