Polynomial super representations of $$U_{q}^{\mathrm{res}}(\mathfrak {gl}_{m|n})$$ at roots of unity

Abstract

As a homomorphic image of the hyperalgebra $$U_{q,R}(m|n)$$ associated with the quantum linear supergroup $$U_{\varvec{\upsilon }}(\mathfrak {gl}_{m|n})$$, 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 field 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 (Brundan and Kujawa in J Algebraic Combin 18:13–39, 2003) 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 Brundan (Proc Lond Math Soc 77:551–581, 1998) for a proof without using the super theory.