Stanley-Reisner rings for symmetric simplicial complexes, G-semimatroids and Abelian arrangements
Alessio D'Ali and Emanuele Delucchi
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Submission date: 23. Apr. 2018
MSC-Numbers: 13F55, 55U10, 06A11
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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.