Suppose the symmetry group $S_n$ acts as a reflection group over the polynomial ring $k.[x_1, \ldots, x_n]$, $k$ are fields such that Char$(k)$ does not divide $n!$. Construct the matrix factorization of this group action discriminant using high-spectrum polynomials. These matrix factorizations are indexed by partitions of $n$ and respect the decomposition of covariate algebras into isomorphic components. The maximal Cohen-Macaulay modules associated with these matrix factorizations give rise to non-commutative solutions of the discriminants and correspond to non-trivial irreducible representations of $S_n$. All structures are implemented in Macaulay2 and some examples are given. We also discuss the extension of these results to the Young subgroup of $S_n$.

    Source link


    Leave A Reply