The multivariate arithmetic Tutte polynomial

Brändén, Petter; Moci, Luca

doi:10.1090/s0002-9947-2014-06092-3

Cited by 49 publications

(100 citation statements)

References 16 publications

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“…The most natural combinatorial structure to consider in this context is of course the poset of layers C, both because this is the direct counterpart of the intersection poset of a hyperplane arrangement and because we already know it determines the Betti numbers and hence (by torsion-freeness) the cohomology groups. As an additional element of similarity, we prove in §7.1 that just as in the case of hyperplane arrangements the cohomology groups can be obtained as the Whitney homology of C. When the arrangement is centered (i.e., defined by kernels of characters), another associated structure is the arithmetic matroid of the defining characters [4]. While for hyperplane arrangements the two counterparts -(semi)lattice of flats and (semi)matroid -are equivalent combinatorial structures, in our situation it is still true that in the centered case C determines an arithmetic matroid, but it is not known at present how to construct C from an abstract arithmetic matroid.…”

Section: Combinatorial Aspectsmentioning

confidence: 95%

“…One such direction, from algebraic combinatorics, led Moci [25] to introduce a suitable generalization of the Tutte polynomials and then, jointly with d'Adderio [8], to the development of arithmetic matroids (for an up-to date account see Brändén and Moci [4]). 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%

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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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Section: Combinatorial Aspectsmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

The integer cohomology algebra of toric arrangements

Callegaro

Delucchi

2017

Advances in Mathematics

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“…Theorem 3.22 implies that the universal Tutte character satisfies a universal convolution formula in the polynomial algebra K[u 0 , v 0 , w 0 , u 1 , v 1 , w 1 , u 2 , v 2 , w 2 ]. One concludes by specializing the variables to (u 0 , v 0 , w 0 , u 1 , v 1 , w 1 , u 2 , v 2 , w 2 ) = (1, −cd, 1, −a, d, −e, −ab, 1, −ef ) and using (10).…”

Section: Delta-matroids and Perspectivesmentioning

confidence: 99%

“…Arithmetic matroids [20,10] are one of several recently introduced kinds of decorated matroid. Matroids encode information about the topology of hyperplane arrangements: notably the graded dimension of the cohomology of a complex hyperplane arrangement complement is an evaluation of the Tutte polynomial.…”

Section: Arithmetic Matroidsmentioning

confidence: 99%

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Universal Tutte characters via combinatorial coalgebras

Dupont¹,

Fink²,

Moci³

2018

Algebraic Combinatorics

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This work discusses the extraction of meaningful invariants of combinatorial objects from coalgebra or bialgebra structures. The Tutte polynomial is an invariant of graphs well known for the formula which computes it recursively by deleting and contracting edges, and for its universality with respect to similar recurrence. We generalize this to all classes of combinatorial objects with deletion and contraction operations, associating to each such class a universal Tutte character by a functorial procedure. We show that these invariants satisfy a universal property and convolution formulae similar to the Tutte polynomial. With this machinery we recover classical invariants for delta-matroids, matroid perspectives, relative and colored matroids, generalized permutohedra, and arithmetic matroids. We also produce some new invariants along with new convolution formulae.obtain several convolution formulae for different classes of combinatorial objects. Some of these formulae are well known, while others are (to the best of our knowledge) new: see for instance Propositions 5.12, 5.13 and 5.14 for the classical Tutte polynomial, Proposition 6.17 for the Las Vergnas polynomial, Proposition 6.20 for the Bollobás-Riordan polynomial and Theorem 10.9 for the arithmetic Tutte polynomial.We have chosen to exemplify our results with mostly matroid-like combinatorial structures, which are arguably simpler to deal with than topological examples. Our formalism, and in particular the mechanism of twist maps, should help uncover new invariants in the latter class and produce interesting convolution formulae. This will be the subject of a subsequent article.Layout. The structure of this paper is as follows. In Section 2 we give the definitions of minors systems and comonoids in set species. Section 3 contains the main results, including the definitions and statements of our universal invariants and formulae. Section 4 presents a number of further results which are less essential for our main developments.The remainder of the paper, Sections 5 through 10, comprises a sequence of applications of our theory to numerous individual minors systems. We work out Grothendieck groups, universal Tutte characters, and in many cases universal convolution formulae, and point out how these specialize to invariants and formulae present in the literature. Section 5 covers the minors system of matroids, of which all the subsequent sections are in one way or another generalizations, together with the minors system of graphs. Of the following sections, Section 6 is on delta-matroids, matroid perspectives, and their ilk is called on in Section 7 on relative matroids, but there are no (or at most incidental) dependences between these and Section 8 on polymatroids and generalised permutohedra, Section 9 on colored matroids, or Section 10 on arithmetic matroids, nor among the latter three, so the reader should have no trouble taking these in any order. The last four sections are also new by comparison with [37].Notation. We fix a commutative ring with unit K...

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Matroids Over a Ring

Fink

Moci

2015

Springer INdAM Series

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The multivariate arithmetic Tutte polynomial

Cited by 49 publications

References 16 publications

The integer cohomology algebra of toric arrangements

The integer cohomology algebra of toric arrangements

Universal Tutte characters via combinatorial coalgebras

Matroids Over a Ring

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