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

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.