Abstract: We give a weight-preserving, sign-reversing involution method for computing the multiplicities of irreducible representations in a certain class of $G \wr S_n$-modules, where $G$ is a finite, abelian group. In general, it is sufficient to consider the case where $G = Z_k$, the cyclic group of order $k$. We do this by "coloring" cycles according to a "color rule", and show that irreducible characters can be enumerated by certain classes of semistandard Young tableaux in the colors.

In this talk we will briefly describe our methods and applications, and focus on an application with geometric motivations, namely, the character of affine semigroup algebras coming from the product of projective toric varieties. In particular, we will give a combinatorial rule for the resulting character, and in the case of a product of simplices (coming from a product of projective spaces), we will derive as a consequence "equivariant" Euler-Mahonian identities. Finally, we mention links to a "refined" version of equivariant Ehrhart theory.

(Joint with Marino Romero, arXiv:2504.19008.)