Positive combinatorial formulae for involution matrix loci and orbit harmonics

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 $a\ne a^{\prime }$ and $d$. Our positive formulae also yield potential attempts to find a linear basis for $R({\mathcal{M}}_{n,a})$ and a statistic $\mathrm{stat}:{\mathcal{M}}_{n,a}\to {\mathbb{Z}}_{\ge 0}$ to interpret the Hilbert series $\mathrm{Hilb}(R({\mathcal{M}}_{n,a});q)$ of $R({\mathcal{M}}_{n,a})$.

14 pages