论文标题
$ s_n $的判别矩阵因法化
Matrix Factorizations of the discriminant of $S_n$
论文作者
论文摘要
考虑对称组$ s_n $作为多项式环$ k [x_1，\ ldots，x_n] $的反射组，其中$ k $是一个字段，以至于char $（k）$不划分$ n！$。我们使用较高的SpecHT多项式来构建该组作用的判别性矩阵因子化：这些矩阵因子化被$ n $的分区索引，并尊重共依体代数分解为同种型组件。与这些矩阵因法化相关的最大Cohen-Macaulay模块会导致判别物的非交通分辨率，它们对应于$ s_n $的非繁琐不可减至的表示。我们所有的结构均在Macaulay2中实现，我们提供了几个示例。我们还讨论了这些结果的扩展，以$ s_n $的年轻子组。
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 $S_n$.
