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

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 $m=n\ge r$ 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.

36 pages