Some Multiplication Formulas in Queer q-Schur Superalgebras

Building on the work (Du and Wan J. Aust. Math. Soc. 105:1–31, 02 2018), where some natural basis for the queer q-Schur superalgebra Qq(n,r;R) is defined by a labelling set of matrices and their associated double coset representatives, we investigate the matrix representation of the regular module of Qq(n,r;R) with respect to this basis. More precisely, we derive explicitly (resp., partially explicitly) the multiplication formulas of the basis elements by certain even (resp., odd) generators of a queer q-Schur superalgebra. These multiplication formulas are highly technical to derive, especially in the odd case. It requires to discover many multiplication (or commutation) formulas in the Hecke–Clifford algebra Hr,Rc associated with the labelling matrices. For example, for a given such a labelling matrix A⋆, there are several matrices w(A), σ(A),A~, and A^ associated with the base matrix A of A⋆, where w(A) is used to compute a reduced expression of the distinguished double coset representatives dA, and the other matrices are used to describe the permutation dA and the SDP (commutation) condition between TdA and generators of the Clifford subsuperalgebra. With these multiplication formulas, we will construct a new realisation of the quantum queer supergroup in a forthcoming paper (Du et al. [13]), and to give new applications to the integral Schur–Weyl–Olshanski duality and its associated representation theory at roots of unity.