A general framework for the polynomiality property of the structure coefficients of double-class algebras

Abstract

In this paper, we build a general framework in which we give a formula that shows the form of the structure coefficients of double-class algebras and centers of group algebras. This formula allows us to give a polynomiality property for the structure coefficients of some important algebras. In particular, we re-obtain the polynomiality property of the structure coefficients in the cases of the center of the symmetric group algebra and the Hecke algebra of the pair $$(\mathcal {S}_{2n},\mathcal {B}_{n}).$$(S2n,Bn). We also assign a new polynomiality property for the structure coefficients of the center of the hyperoctahedral group algebra and the Hecke algebra of the pair $$(\mathcal {S}_n\times \mathcal {S}_{n-1}^\mathrm{opp},{{\mathrm{diag}}}(\mathcal {S}_{n-1}))$$(Sn×Sn-1opp,diag(Sn-1)).