Generalized coinvariant algebras for wreath products

Abstract

Let r be a positive integer and let Gn be the reflection group of n×n monomial matrices whose entries are rth complex roots of unity and let k≤n. We define and study two new graded quotients Rn,k and Sn,k of the polynomial ring C[x1,…,xn] in n variables. When k=n, both of these quotients coincide with the classical coinvariant algebra attached to Gn. The algebraic properties of our quotients are governed by the combinatorial properties of k-dimensional faces in the Coxeter complex attached to Gn (in the case of Rn,k) and r-colored ordered set partitions of {1,2,…,n} with k blocks (in the case of Sn,k). Our work generalizes a construction of Haglund, Rhoades, and Shimozono from the symmetric group Sn to the more general wreath products Gn.