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 $G_n.$ 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 $G_n.$ 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 $(\mathcal{S}_{2n},\mathcal{B}_{n}).$ We also give a new polynomiality property for the structure coefficients of the center of the hyperoctahedral group algebra and the double-class algebra $\mathbb{C}[diag(\mathcal{S}_{n-1})\setminus \mathcal{S}_n\times \mathcal{S}^{opp}_{n-1}/ diag(\mathcal{S}_{n-1})].$