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.

中文翻译：

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$$ U_ {q} ^ {\ mathrm {res}}（\ mathfrak {gl} _ {m | n}）$$ Uqres（glm | n）的多项式超级表示形式

作为与量子线性超群\（U _ {\ varvec {\ upsilon}}（\ mathfrak {gl} _ {m | n}相关的超代数\ { U_ {q，R}（m | n）\）的同态图像}）\），我们首先在交换环R上给出q -Schur超代数\（S_ {q，R}（m | n，r）\）的表示。然后，我们为字段F上\（U_ {q，F}（m | n）\）的多项式超模制定一个准则，并使用该准则确定单位根处的多项式不可约超模的分类。这也给出了所有r的不可约\（S_ {q，F}（m | n，r）\）-超模的分类。当\（m = n \ ge r \）时作为应用程序受经典著作（非量子）中优美的作品（Brundan和Kujawa在J Algebraic Combin 18：13–39，2003）的启发，我们为与Hecke代数上不可约模有关的Mullineux猜想提供了新的证明\（H_ {q ^ 2，F}（{{\ mathfrak {S}}} _ r）\） ; 请参见Brundan（Proc Lond Math Soc 77：551–581，1998年）获取不使用超理论的证明。