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arXiv · 2308.02112

Some multiplication formulas in queer $q$-Schur superalgebras

Jie Du·Haixia Gu·Zhenhua Li·Jinkui Wan
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Abstract

Building on the work [18], where some standard basis for the queer $q$-Schur superalgebra $\mathcal{Q}_q(n,r;R)$ is defined by a labelling set of matrices and their associated double coset representatives, we investigate the matrix representation of the regular module of $\mathcal{Q}_q(n,r;R)$ with respect to this basis. More precisely, we derive explicitly (resp., partial explicitly) the multiplication formulas of the basis elements by certain even (resp., odd) generators of a queer $q$-Schur superalgebra. These multiplication formulas are highly technical to derive, especially in the odd case. It requires to discover many multiplication (or commutation) formulas in the Hecke--Clifford algebra $\mathcal{H}_{r,R}^c$ associated with the labelling matrices. For example, for a given such a labelling matrix $A^{\!\star}$, there are several matrices $w(A)$, $σ(A), \widetilde A$, and $\widehat A$ associated with the base matrix $A$ of $A^{\!\star}$, where $w(A)$ is used to compute a reduced expression of the distinguished double coset representatives $d_A$, and the other matrices are used to describe the permutation $d_A$ and the SDP (commutation) condition between $T_{d_A}$ and generators of the Clifford subsuperalgebra. With these multiplication formulas, we will construct a new realisation of the quantum queer supergroup in a forthcoming paper [13], and to give new applications to the integral Schur--Olshanski duality and its associated representation theory at roots of unity.

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BibTeXRIS

Jie Du, Haixia Gu, Zhenhua Li, Jinkui Wan. 2023-08-04. Some multiplication formulas in queer $q$-Schur superalgebras. https://arxiv.org/abs/2308.02112

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