Involution matrix loci and orbit harmonics

TL;DR

This paper introduces new graded quotient rings derived from matrix loci associated with involutions in the symmetric group, revealing connections to Tracy-Widom distributions and plethysm in symmetric functions.

Contribution

It constructs and analyzes quotient rings from matrix loci related to involutions, linking algebraic geometry, combinatorics, and probability in novel ways.

Findings

Hilbert series related to Tracy-Widom distributions

Refinement of plethysm $s_{n/2}[s_2]$ via graded Frobenius image

Action of symmetric group on quotient rings

Abstract

Let $\mathrm{Mat}_{n \times n}(\mathbb{C})$ be the affine space of $n \times n$ complex matrices with coordinate ring $\mathbb{C}[\mathbf{x}_{n \times n}]$. We define graded quotients of $\mathbb{C}[\mathbf{x}_{n \times n}]$ which carry an action of the symmetric group $\mathfrak{S}_n$ by simultaneous permutation of rows and columns. These quotient rings are obtained by applying the orbit harmonics method to matrix loci corresponding to all involutions in $\mathfrak{S}_n$ and the conjugacy classes of involutions in $\mathfrak{S}_n$ with a given number of fixed points. In the case of perfect matchings on $\{1, \dots, n\}$ with $n$ even, the Hilbert series of our quotient ring is related to Tracy-Widom distributions and its graded Frobenius image gives a refinement of the plethysm $s_{n/2}[s_2]$.