Jack polynomials and the coinvariant ring of $G(r,p,n)$

TL;DR

This paper explores the structure of the coinvariant ring of complex reflection groups G(r,p,n) using non-symmetric Jack polynomials, revealing its decomposition into irreducible modules for a related algebra.

Contribution

It constructs a basis of non-symmetric Jack polynomials for the coinvariant ring and decomposes it into irreducible modules, linking descent monomials to colored descent representations.

Findings

Basis of non-symmetric Jack polynomials for coinvariant ring

Decomposition into irreducible modules for the rational Cherednik algebra

Connection between descent monomials and colored descent representations

Abstract

We study the coinvariant ring of the complex reflection group $G(r,p,n)$ as a module for the corresponding rational Cherednik algebra $\HH$ and its generalized graded affine Hecke subalgebra $\mathcal{H}$. We construct a basis consisting of non-symmetric Jack polynomials, and using this basis decompose the coinvariant ring into irreducible modules for $\mathcal{H}$. The basis consists of certain non-symmetric Jack polynomials, whose leading terms are the ``descent monomials'' for $G(r,p,n)$ recently studied by Adin, Brenti, and Roichman and Bagno and Biagoli. The irreducible $\mathcal{H}$-submodules of the coinvariant ring are their ``colored descent representations''.