Involution matrix loci and orbit harmonics

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 $s_{n/2}[s_2]$.

33 pages