The algebra of conjugacy classes of the wreath product of a finite group with the symmetric group

TL;DR

This paper introduces a new combinatorial algebraic framework to analyze the structure coefficients of the center of wreath product groups, proving they are polynomials in the group size with non-negative coefficients.

Contribution

It generalizes Ivanov and Kerov's method to wreath products, establishing polynomiality of structure coefficients in a broad algebraic setting.

Findings

Structure coefficients are polynomials in n with non-negative integer coefficients.

Introduces the concept of G-partial permutations for wreath product analysis.

Extends polynomiality results from symmetric groups to wreath products.

Abstract

For a finite group $G,$ we define the concept of $G$-partial permutation and use it to show that the structure coefficients of the center of the wreath product $G\wr \mathcal{S}_n$ algebra are polynomials in $n$ with non-negative integer coefficients. Our main tool is a combinatorial algebra which projects onto the center of the group $G\wr \mathcal{S}_n$ algebra for every $n.$ This generalizes the Ivanov and Kerov method to prove the polynomiality property for the structure coefficients of the center of the symmetric group algebra.