Multiplication formulas and semisimplicity for q-Schur superalgebras

TL;DR

This paper develops multiplication formulas for q-Schur superalgebras using combinatorial methods, leading to criteria for their semisimplicity and constructions of related algebraic structures.

Contribution

It introduces new multiplication formulas for q-Schur superalgebras based on combinatorial analysis of symmetric group cosets, advancing the understanding of their structure.

Findings

Derived multiplication formulas for q-Schur superalgebras

Established semisimplicity criteria for these algebras

Constructed infinitesimal and little q-Schur superalgebras

Abstract

We investigate products of certain double cosets for the symmetric group and use the findings to derive some multiplication formulas for q-Schur superalgebras. This gives a combinatorialisation of the relative norm approach developed by the first two authors. 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.