Positive combinatorial formulae for involution matrix loci and orbit harmonics
Hai Zhu
TL;DR
This work advances the orbit-harmonics study of the involution matrix locus ${\mathcal{M}}_{n,a}$ by converting a previously signed ${\mathrm{grFrob}}(R({\mathcal{M}}_{n,a});q)$ into positive combinatorial formulas. Using lattice-path constructions, horizontal stripes, and shadow maps, the authors first derive a positive, component-dependent expression, then upgrade it to an explicit graded refinement that aligns with the Schur expansion of $h_{(n-a)/2}[h_2]\cdot h_a$ through a bijection between index sets. The resulting formula expresses ${\mathrm{grFrob}}(R({\mathcal{M}}_{n,a});q)$ as a sum of Schur functions over width-constrained horizontal stripes, yielding ${\mathfrak{S}}_n$-equivariant connections between graded components and suggesting avenues for a basis and a statistic interpreting the Hilbert series. These refinements facilitate potential basis constructions and deeper combinatorial interpretations, linking involution loci, symmetric functions, and representation theory in a concrete, positive framework. The work also points to further directions, including explicit statistics and monomial bases, and highlights applications to equivariant isomorphisms across different fixed-point counts.
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 $R(\mathcal{M}_{n,a})$.