Positive combinatorial formulae for involution matrix loci and orbit harmonics

TL;DR

This paper develops positive combinatorial formulas for the graded Frobenius image of modules associated with involutions in symmetric groups, enabling better understanding of their structure and potential bases.

Contribution

It provides two positive combinatorial formulas for the Frobenius image of involution matrix loci modules, improving upon previous signed formulas.

Findings

Derived positive formulas for the Frobenius image

Established isomorphisms between graded components for different fixed points

Suggested potential bases and statistics for these modules

Abstract

Let $\mathcal{M}_{n,a}$ be the set consisting of involutions in symmetric group $\mathfrak{S}_n$ with exactly $a$ fixed points and apply the orbit harmonics method to obtain a graded $\mathfrak{S}_n$-module $R(\mathcal{M}_{n,a})$. Liu, Ma, Rhoades, and Zhu figured out a signed combinatorial formula for the graded Frobenius image $\mathrm{grFrob}(R(\mathcal{M}_{n,a});q)$ of $R(\mathcal{M}_{n,a})$. Our goal is to cancel these signs. Finally, we find two positive combinatorial formulae for $\mathrm{grFrob}(R(\mathcal{M}_{n,a});q)$. As an application, we deduce a series of $\mathfrak{S}_n$-equivariant isomorphisms between graded components $R(\mathcal{M}_{n,a})_d$ and $R(\mathcal{M}_{n,a^{\prime}})_d$ for some integers $a\neq a^{\prime}$ and $d$. Our positive formulae also yield potential attempts to find a linear basis for $R(\mathcal{M}_{n,a})$ and a statistic $\mathrm{stat}:\mathcal{M}_{n,a}\rightarrow\mathbb{Z}_{\ge0}$ to interpret the Hilbert series $\mathrm{Hilb}(R(\mathcal{M}_{n,a});q)$ of $R(\mathcal{M}_{n,a})$.