Descent Representations of Generalized Coinvariant Algebras

TL;DR

This paper extends the combinatorial and algebraic understanding of coinvariant algebras and their generalizations for symmetric groups and complex reflection groups, connecting to the Delta Conjecture and Macdonald polynomials.

Contribution

It generalizes the descent representation descriptions from classical coinvariant algebras to their extended versions for complex reflection groups.

Findings

Extended the Frobenius image description to $R_{n,k}$ and $S_{n,k}$

Connected descent representations to the Delta Conjecture

Provided new combinatorial bases for generalized coinvariant algebras

Abstract

The coinvariant algebra $R_n$ is a well-studied $\mathfrak{S}_n$-module that is a graded version of the regular representation of $\mathfrak{S}_n$. Using a straightening algorithm on monomials and the Garsia-Stanton basis, Adin, Brenti, and Roichman gave a description of the Frobenius image of $R_n$, graded by partitions, in terms of descents of standard Young tableaux. Motivated by the Delta Conjecture of Macdonald polynomials, Haglund, Rhoades, and Shimozono gave an extension of the coinvariant algebra $R_{n,k}$ and an extension of the Garsia-Stanton basis. Chan and Rhoades further extend these results from $\mathfrak{S}_n$ to the complex reflection group $G(r,1,n)$ by defining a $G(r,1,n)$ module $S_{n,k}$ that generalizes the coinvariant algebra for $G(r,1,n)$. We extend the results of Adin, Brenti, and Roichman to $R_{n,k}$ and $S_{n,k}$.