Stanley-Reisner rings for quasi-arithmetic matroids

Lenz, Matthias

doi:10.48550/arxiv.1709.03834

Cited by 1 publication

(4 citation statements)

References 27 publications

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“…In the special case of hyperplane arrangements, we recover the classical theory of matroids and their Stanley-Reisner rings. In the case of (central) toric arrangements our rings are isomorphic to constructions that appeared in previous literature [36,39], and the action's Tutte polynomial is the arithmetic Tutte polynomial [43]. We obtain similar results also in the even broader context given by .p; q/arrangements, part of a class of arrangements in Abelian Lie groups studied by Liu, Tran and Yoshinaga [37].…”

Section: Application: Abelian Arrangementssupporting

confidence: 72%

“…Remark 9.5. Notice that when A is central and toric, via the case k D 1 of Lemma 9.2 we recover the ring of [36,39], where item (iv) of Theorem 6 is proved in the corresponding situation.…”

Section: Remark 93 Ifmentioning

confidence: 68%

“…G. Any choice of a 1 ; : : : ; a n 2 ƒ and b 1 ; : : : ; b n 2 G determines an arrangement A central arrangement is one where b i D id G for all i D 1; : : : ; n. A deep enumerative-combinatorial study of central arrangements, with special attention to the linear and toric case, has led to the introduction of arithmetic Tutte polynomials [43] and arithmetic matroids [14,23]. Questions about commutativealgebraic interpretations of some of the polynomials arising in this context led to attempts at modeling the poset I.A/ of "independent sets" in the linear and (central) toric case, defined to be the set of pairs .X; c/ where X is a (Q-)linearly independent subset of fa i g i and c is a connected component of the intersection of the corresponding hypersurfaces [36,39].1…”

Section: Abelian Arrangementsmentioning

confidence: 99%

“…1The definitions in [36,39] are formally in terms of pairs .X; g/ where g is a torsion element of the quotient group ƒ=hX i Z . Such torsion elements are however in (natural) bijection with the connected components of the intersection of the hypersurfaces determined by the elements of X (see, e.g., [28,43]).…”

Section: Abelian Arrangementsmentioning

confidence: 99%

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

D’Alì

Delucchi

2021

J. Comb. Algebra

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We extend the notion of face rings of simplicial complexes and simplicial posets to the case of finite-length (possibly infinite) 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 under a mild condition on the group action. We also identify a class of actions on simplicial complexes that preserve the homotopical Cohen-Macaulay property under quotients.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 group action. Moreover, in this case an additional condition on the action ensures that the quotient poset is Cohen-Macaulay in characteristic 0 and every characteristic that does not divide an explicitly computable number. This implies the same property for the associated Stanley-Reisner rings. In particular, this holds for independence posets and rings associated to toric, elliptic and, more generally, .p; q/-arrangements.As a byproduct, we prove that posets of connected components (also known as posets of layers) of such arrangements are Cohen-Macaulay with the same condition on the characteristic.

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Section: Application: Abelian Arrangementssupporting

confidence: 72%

“…Remark 9.5. Notice that when A is central and toric, via the case k D 1 of Lemma 9.2 we recover the ring of [36,39], where item (iv) of Theorem 6 is proved in the corresponding situation.…”

Section: Remark 93 Ifmentioning

confidence: 68%