Stanley–Reisner rings for symmetric simplicial complexes, $G$-semimatroids and Abelian arrangements

Pub Date : 2018-04-19 DOI:10.4171/JCA/53
Alessio D'Alì, Emanuele Delucchi
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引用次数: 7

Abstract

We extend the notion of face rings of simplicial complexes and simplicial posets to the case of finite-length simplicial posets with a group action. The action on the complex induces an action on the face ring, and we prove that the ring of invariants is isomorphic to the face ring of the quotient simplicial poset when the group action is translative (in the sense of Delucchi-Riedel). When the acted-upon poset is the independence complex of a semimatroid, the h-polynomial of the ring of invariants can be read off the Tutte polynomial of the associated G-semimatroid. We thus recover the classical theory in the case of trivial group actions on finite simplicial posets and, in the special case of central toric arrangements, our rings are isomorphic to those defined by Martino and by Lenz. We also describe a further condition on the group action ensuring that the topological Cohen-Macaulay property is preserved under quotients. In particular, we prove that the independence complex and the Stanley-Reisner ring of any Abelian arrangement are Cohen-Macaulay over every field. As a byproduct, we prove that posets of connected components (also known as posets of layers) of Abelian arrangements are (homotopically) Cohen-Macaulay.
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对称单复形、$G$-半拟阵和阿贝尔排列的Stanley–Reisner环
我们将单纯复形和单纯偏序集的面环的概念推广到具有群作用的有限长单纯偏序集合的情况。复形上的作用诱导了面环上的作用,我们证明了当群作用是平移的(在Delucchi-Riedel的意义上)时,不变量环同构于商单偏序集的面环。当作用偏序集是半拟阵的独立复形时,不变量环的h多项式可以从相关的G-半拟阵中的Tutte多项式中读出。因此,我们在有限单纯偏序集上的平凡群作用的情况下恢复了经典理论,并且在中心复曲面排列的特殊情况下,我们的环同构于Martino和Lenz定义的环。我们还描述了群作用的另一个条件,它确保拓扑Cohen—Macaulay性质在商下保持。特别地,我们证明了任何Abelian排列的独立复形和Stanley-Resner环在每个域上都是Cohen-Macaulay。作为副产品,我们证明了阿贝尔排列的连通分量的偏序集(也称为层的偏序集合)是(同源的)Cohen Macaulay。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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