We investigate products of certain double cosets for the symmetric group and use the findings to derive some multiplication formulas for the $q$-Schur superalgebras. This gives a combinatorialization of the relative norm approach developed in Du and Gu (A realization of the quantum supergroup$\mathbf{U}(\mathfrak{g}\mathfrak{l}_{m|n})$, J. Algebra 404 (2014), 60–99). We then give several applications of the multiplication formulas, including the matrix representation of the regular representation and a semisimplicity criterion for $q$-Schur superalgebras. We also construct infinitesimal and little $q$-Schur superalgebras directly from the multiplication formulas and develop their semisimplicity criteria.

中文翻译：

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-SCHUR 超代数的乘法公式和半单性

我们研究对称群的某些双陪集的乘积，并使用这些发现推导出一些乘法公式$q$-舒尔超代数。这给出了 Du 和 Gu 开发的相对规范方法的组合化（量子超群的实现$\mathbf{U}(\mathfrak{g}\mathfrak{l}_{m|n})$, J. 代数404（2014），60-99）。然后我们给出了乘法公式的几种应用，包括正则表示的矩阵表示和$q$-舒尔超代数。我们还构造了无穷小和小$q$-Schur 超代数直接从乘法公式中发展出他们的半简单性准则。