Polynomial super representations of the hyperalgebra of $\mathfrak{gl}_{m|n}$ at roots of unity

TL;DR

This paper develops a framework for understanding polynomial supermodules of the quantum general linear supergroup at roots of unity, classifies irreducible supermodules, and provides a new proof of the Mullineux conjecture in the super setting.

Contribution

It introduces a presentation for the $q$-Schur superalgebra, classifies polynomial irreducible supermodules at roots of unity, and offers a new proof of the Mullineux conjecture in the super context.

Findings

Classification of polynomial irreducible supermodules at roots of unity.

Presentation of the $q$-Schur superalgebra over a commutative ring.

A new proof of the Mullineux conjecture for supermodules.

Abstract

As a homomorphic image of the hyperalgebra $U_{q,R}(m|n)$ associated with the quantum linear supergroup $U_\upsilon(\mathfrak{gl}_{m|n})$, we first give a presentation for the $q$-Schur superalgebra $S_{q,R}(m|n,r)$ over a commutative ring $R$. We then develop a criterion for polynomial supermodules of $U_{q,F}(m|n)$ over a filed $F$ and use this to determine a classification of polynomial irreducible supermodules at roots of unity. This also gives classifications of irreducible $S_{q,F}(m|n,r)$-supermodules for all $r$. As an application when $m=n\geq r$ and motivated by the beautiful work \cite{bru} in the classical (non-quantum) case, we provide a new proof for the Mullineux conjecture related to the irreducible modules over the Hecke algebra $H_{q^2,F}({\mathfrak S}_r)$; see \cite{Br} for a proof without using the super theory.