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 \(\mathcal {Q}_{q}(n,r; R)\) is defined by a labelling set of matrices and their associated double coset representatives, we investigate the matrix representation of the regular module of \(\mathcal {Q}_{q}(n,r; R)\) 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 \(\mathcal {H}^c_{r, R}\) associated with the labelling matrices. For example, for a given such a labelling matrix \({A^{\!\star }}\) , there are several matrices w(A), \(\sigma (A), \widetilde{A}\) , and \(\widehat{A}\) associated with the base matrix A of \({A^{\!\star }}\) , where w(A) is used to compute a reduced expression of the distinguished double coset representatives \(d_A\) , and the other matrices are used to describe the permutation \(d_A\) and the SDP (commutation) condition between \(T_{d_A}\) 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.
