A charge monomial basis of the Garsia-Procesi ring
Mitsuki Hanada
TL;DR
This work constructs a charge-word–based monomial basis for the Garsia-Procesi ring $R_ u$ by imposing catabolizability-type conditions on insertion tableaux. The basis is shown to coincide with the Carlsson-Chou descent basis, linking the algebraic structure of $R_ u$ directly to the catabolism formula for modified Hall-Littlewood polynomials $ ilde{H}_{ u}[X;q]$ and providing an elementary proof that the graded Frobenius character equals $ ilde{H}_{ u}[X;q]$. The paper then leverages this basis to give a transparent description of antisymmetric components under Young subgroups, yielding explicit bases and Hilbert series that recover the same Frobenius character and Hilbert data. Overall, the results connect tableau catabolism, descent- and charge-based bases, and Hall-Littlewood combinatorics to illuminate the structure of $R_ u$ and its symmetric-function character.
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 $\tilde{H}_μ[X;q]$, 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 $R_μ$ with respect to a Young subgroup $S_γ$ 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 $R_μ$ is given by the catabolizability formula for $\tilde{H}_μ[X;q]$.