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THE QUEER
$q$-SCHUR SUPERALGEBRA
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 105 / Issue 3 / December 2018
- Published online by Cambridge University Press:
- 02 February 2018, pp. 316-346
- Print publication:
- December 2018
-
- Article
-
- You have access
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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.
