A charge monomial basis of the Garsia-Procesi ring

TL;DR

This paper introduces a new basis for the Garsia-Procesi ring using combinatorial statistics from Young tableaux, linking it to Hall-Littlewood polynomials and providing an elementary proof of related character formulas.

Contribution

It constructs a charge monomial basis connecting tableau combinatorics with Hall-Littlewood polynomials, and relates antisymmetric parts to tableau conditions, offering new insights into the ring's structure.

Findings

Established a basis equal to the descent basis of Carlsson-Chou (2024+)

Connected the basis to the combinatorial formula for modified Hall-Littlewood polynomials

Provided an elementary proof for the graded Frobenius character of the ring

Abstract

We construct a basis of the Garsia-Procesi ring using the catabolizability type of standard Young tableaux and the charge statistic. This basis turns out to be equal to the descent basis defined in Carlsson-Chou (2024+). Our new construction connects the combinatorics of the basis with the well-known combinatorial formula for the modified Hall-Littlewood polynomials $\tilde{H}_\mu[X;q]$, due to Lascoux, which expresses the polynomials as a sum over standard tableaux that satisfy a catabolizability condition. In addition, we prove that identifying a basis for the antisymmetric part of $R_{\mu}$ with respect to a Young subgroup $S_\gamma$ is equivalent to finding pairs of standard tableaux that satisfy conditions regarding catabolizability and descents. This gives an elementary proof of the fact that the graded Frobenius character of $R_{\mu}$ is given by the catabolizability formula for $\tilde{H}_\mu[X;q]$.