Matrix factorizations of the discriminant of Sn

Abstract

Consider the symmetric group Sn acting as a reflection group on the polynomial ring k[x1,…,xn] 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 Sn. All our constructions are implemented in Macaulay2 and we provide several examples. We also discuss extensions of these results to Young subgroups of Sn and indicate how to generalize them to the reflection groups G(m,1,n).