Let \(k\) be a field and let \(f \in k[x]\). The classical discriminant \(D(f)\) of \(f\) detects whether \(f\) has a multiple root. In general, an explicit formula for \(D(f)\) consists of many monomial terms and several compact determinantal formulae are known, that is, \(D(f)\) can be written as determinant of a matrix with entries polynomials in the coefficients of \(f\).
In this paper, the authors interpret the classical discriminant as the discriminant of the reflection group \(S_{n}\) acting on \(k^{n}\). Let \(G \leq \mathrm{GL}_{n}(k)\) be any finite reflection group acting on the vector space \(k^{n}\), then \(G\) also acts on \(S=\mathrm{Sym}_{k}(k^{n})\). Let \(R=S^{G}\) be the invariant ring under the group action, \(\mathcal{A}(G)\) the reflection arrangement in \(k^{n}\) and \(V(\Delta)\) the discriminant in the (smooth) quotient space \(k^{n}/G\). The hypersurface \(V(\Delta)\) is given by the reduced polynomial \(\Delta \in R\) and is the projection of \(\mathcal{A}(G)\) onto the quotient. Moreover, if \(G=S_{n}\) and \(k=\mathbb{C}\), it is well-known that \(V(\Delta)\) is isomorphic to the classical discriminant \(V(D (f))\), where \(f\) is a polynomial of degree \(n\).
In this paper, the authors 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 non-commutative resolution of the discriminant and they correspond to the nontrivial irreducible representations of \(S_{n}\). All such constructions are implemented in Macaulay2. The authors also discuss extensions of these results to Young subgroups of \(S_{n}\) and they indicate how to generalize them to the reflection groups \(G(m,1,n)\).