Matrix factorizations of the discriminant of Sn

Abstract

Consider the symmetric group $S_n$ acting as a reflection group on the polynomial ring $k[x_1,\dots ,x_n]$ 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 $S_n$ and indicate how to generalize them to the reflection groups $G(m,1,n)$.