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MULTIPLICATION FORMULAS AND SEMISIMPLICITY FOR $q$ -SCHUR SUPERALGEBRAS

  • JIE DU (a1), HAIXIA GU (a2) and ZHONGGUO ZHOU (a3)

Abstract

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.

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The work was supported by a 2017 UNSW Science Goldstar Grant and the Natural Science Foundation of China (#11501197, #11671234). The third author would like to thank UNSW for its hospitality during his a year visit and thank the Jiangsu Provincial Department of Education for financial support.

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MULTIPLICATION FORMULAS AND SEMISIMPLICITY FOR $q$ -SCHUR SUPERALGEBRAS

  • JIE DU (a1), HAIXIA GU (a2) and ZHONGGUO ZHOU (a3)

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