Color rules for cyclic wreath products and semigroup algebras from projective toric varieties

Abstract

We introduce the notion of "color rules" for computing class functions of $Z_k \wr S_n$, where $Z_k$ is the cyclic group of order $k$ and $S_n$ is the symmetric group on $n$ letters. Using a general sign-reversing involution and a map of order $k$, we give a combinatorial proof that the irreducible decomposition of these class functions is given by a weighted sum over semistandard tableaux in the colors. Since using two colors at once is also a color rule, we are consequently able to decompose arbitrary tensor products of representations whose characters can be computed via color rules. This method extends to class functions of $G \wr S_n$ where $G$ is a finite abelian group. We give a number of applications, including decomposing tensor powers of the defining representation, along with a combinatorial proof of the Murnaghan-Nakayama rule for $Z_k \wr S_n$. Our main application is to the study of the linear action of $Z_k \wr S_n$ on bigraded affine semigroup algebras arising from the product of projective toric varieties. In the case of the product of projective spaces, our methods give the decomposition of these bigraded characters into irreducible characters, thus deriving equivariant generalizations of Euler-Mahonian identities.