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

Take a sequence of couples $(G_n,K_n{)}_n$, where $G_n$ is a group and $K_n$ is a sub-group of $G_n\mathrm{.}$ Under some conditions, we are able to give a formula that shows the form of the structure coefficients that appear in the product of double-classes of $K_n$ in $G_n\mathrm{.}$ We show how this can give us a similar result for the structure coefficients of the centers of group algebras. These formulas allow us to 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 $({\mathcal{S}}_{2n},{\mathcal{B}}_n)\mathrm{.}$ We also give a new polynomiality property for the structure coefficients of the center of the hyperoctahedral group algebra and the double-class algebra $$\mathbb{C}$[diag($\mathcal{S}${n-1})\setminus $\mathcal{S}$n\times $\mathcal{S}$^{opp}{n-1}/ diag($\mathcal{S}${n-1})].$