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

Abstract

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 $\mathbb{C}[diag({\mathcal{S}}_{n-1})\setminus {\mathcal{S}}_n\times {\mathcal{S}}_{n-1}^{opp}/diag({\mathcal{S}}_{n-1})].$

References in corpus (3)

• On the excedance number of colored permutation groups
• A Frobenius formula for the structure coefficients of double-class algebras of Gelfand pairs
• Polynomialité des coefficients de structure des algèbres de doubles-classes

Cited by in corpus (4)

• A Frobenius formula for the structure coefficients of double-class algebras of Gelfand pairs
• $k$-partial permutations and the center of the wreath product ${\mathcal{S}}_k\wr {\mathcal{S}}_n$ algebra
• The center of the wreath product of symmetric groups algebra
• The algebra of conjugacy classes of the wreath product of a finite group with the symmetric group