Matrix Factorizations of the discriminant of $S_n$

TL;DR

This paper constructs matrix factorizations of the discriminant of the symmetric group acting on polynomial rings, using Higher Specht polynomials, leading to noncommutative resolutions linked to irreducible representations.

Contribution

It introduces a novel method to produce matrix factorizations of the discriminant for $S_n$ using Higher Specht polynomials, connecting representation theory with algebraic geometry.

Findings

Matrix factorizations indexed by partitions of n

Noncommutative resolutions of the discriminant

Implementation in Macaulay2 with examples

Abstract

Consider the symmetric group $S_n$ acting as a reflection group on the polynomial ring $k[x_1, \ldots, x_n]$, where $k$ is a field such that Char$(k)$ does not divide $n!$. We use Higher Specht polynomials to construct matrix factorizations of the discriminant of this group action: these matrix factorizations are indexed by partitions of $n$ and respect the decomposition of the coinvariant algebra into isotypical components. The maximal Cohen-Macaulay modules associated to these matrix factorizations give rise to a noncommutative resolution of the discriminant and they correspond to the nontrivial irreducible representations of $S_n$. All our constructions are implemented in Macaulay2 and we provide several examples. We also discuss extensions of these results to Young subgroups of $S_n$.