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arXiv · 2209.03375

Matrix Factorizations of the discriminant of $S_n$

Eleonore Faber·Colin Ingalls·Simon May·Marco Talarico
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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 $S_n$.

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BibTeXRIS

Eleonore Faber, Colin Ingalls, Simon May, Marco Talarico. 2022-09-07. Matrix Factorizations of the discriminant of $S_n$. https://arxiv.org/abs/2209.03375

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