Positive Combinatorial Formulae for Involution Matrix Loci and Orbit Harmonics

Abstract

Let $\mathcal{M}_{n,a}$ be the set consisting of involutions in the 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 $R(\mathcal{M}_{n,a})$. We cancel these minus signs using lattice paths. Finally, we also 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 $R(\mathcal{M}_{n,a})$.