Let \({\textrm{Mat}}_{n \times n}({\mathbb {C}})\) be the affine space of \(n \times n\) complex matrices with coordinate ring \({\mathbb {C}}[{\textbf{x}}_{n \times n}]\) . We define graded quotients of \({\mathbb {C}}[{\textbf{x}}_{n \times n}]\) 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 \(\{1, \ldots , n\}\) 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 \(h_{n/2}[h_2]\) .
