Group actions on semimatroids

Delucchi, Emanuele; Riedel, Sonja

doi:10.1016/j.aam.2017.11.001

Cited by 17 publications

(44 citation statements)

References 35 publications

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“…For an in-depth discussion of this question see [Pag17]. The paper [DR18] introduces group actions on semimatroids as an attempt for a unified axiomatization of posets of layers and multiplicity functions.…”

Section: Toric Arrangementsmentioning

confidence: 99%

Orlik-Solomon type presentations for the cohomology algebra of toric arrangements

Callegaro

D'Adderio

Delucchi

et al. 2019

Trans. Amer. Math. Soc.

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We give an explicit presentation for the integral cohomology ring of the complement of any arrangement of level sets of characters in a complex torus (alias "toric arrangement"). Our description parallels the one given by Orlik and Solomon for arrangements of hyperplanes, and builds on De Concini and Procesi's work on the rational cohomology of unimodular toric arrangements. As a byproduct we extend Dupont's rational formality result to formality over Z.The data needed in order to state the presentation is fully encoded in the poset of connected components of intersections of the arrangement.

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Section: Toric Arrangementsmentioning

confidence: 99%

Orlik-Solomon type presentations for the cohomology algebra of toric arrangements

Callegaro

D'Adderio

Delucchi

et al. 2019

Trans. Amer. Math. Soc.

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“…We will see that many known properties (deletion-contraction formula, Euler characteristic of the complement, point counting, Poincaré polynomial, convolution formula) for (arithmetic) Tutte polynomials are shared by G-Tutte polynomials. (See [17] for another attempt to generalize arithmetic Tutte polynomials. )…”

mentioning

confidence: 99%

G-Tutte Polynomials and Abelian Lie Group Arrangements

Liu

Tran

Yoshinaga

2019

International Mathematics Research Notices

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We introduce and study the notion of the G-Tutte polynomial for a list A of elements in a finitely generated abelian group Γ and an abelian group G, which is defined by counting the number of homomorphisms from associated finite abelian groups to G.The G-Tutte polynomial is a common generalization of the (arithmetic) Tutte polynomial for realizable (arithmetic) matroids, the characteristic quasi-polynomial for integral arrangements, Brändén-Moci's arithmetic version of the partition function of an abelian group-valued Potts model, and the modified Tutte-Krushkal-Renhardy polynomial for a finite CW-complex.As in the classical case, G-Tutte polynomials carry topological and enumerative information (e.g., the Euler characteristic, point counting and the Poincaré polynomial) of abelian Lie group arrangements.We also discuss differences between the arithmetic Tutte and the G-Tutte polynomials related to the axioms for arithmetic matroids and the (non-)positivity of coefficients.

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“…These objects, as well as others like matroids over rings [18], exhibit an interesting structure theory and recover earlier enumerative results by Ehrenborg, Readdy and Slone [17] and Lawrence [21] but, as of yet, only bear an enumerative relationship with topological or geometric invariants of toric arrangements -in particular, these structures do not characterize their intersection pattern (one attempt towards closing this gap has been made by considering group actions on semimatroids [14]). …”

Section: Introductionmentioning

confidence: 99%

“…The two sheaves that appear in (14) are two restrictions of the same constant sheaf, thus the map is the projection…”

Section: The Complexified Casementioning

confidence: 99%

The integer cohomology algebra of toric arrangements

Callegaro

Delucchi

2017

Advances in Mathematics

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Abstract. We compute the cohomology ring of the complement of a toric arrangement with integer coefficients and investigate its dependency from the arrangement's combinatorial data. To this end, we study a morphism of spectral sequences associated to certain combinatorially defined subcomplexes of the toric Salvetti category in the complexified case, and use a technical argument in order to extend the results to full generality. As a byproduct we obtain:-a "combinatorial" version of Brieskorn's lemma in terms of Salvetti complexes of complexified arrangements, -a uniqueness result for realizations of arithmetic matroids with at least one basis of multiplicity 1.

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Group actions on semimatroids

Cited by 17 publications

References 35 publications

Orlik-Solomon type presentations for the cohomology algebra of toric arrangements

Orlik-Solomon type presentations for the cohomology algebra of toric arrangements

G-Tutte Polynomials and Abelian Lie Group Arrangements

The integer cohomology algebra of toric arrangements

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