Arithmetic matroids, the Tutte polynomial and toric arrangements

D'Adderio, Michele; Moci, Luca

doi:10.1016/j.aim.2012.09.001

Cited by 52 publications

(54 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If R is a domain then this means that these two quantities are non-zero. If this condition is satisfied then equations (17), (18) and (19) are equivalent to (20) u…”

Section: Comparison Tomentioning

confidence: 99%

“…We note that equation (20) amounts to saying that N 1 and N 2 are norms. This is another argument for the naturality of our use of norms in our deletion-contraction recurrence formula.…”

Section: Comparison Tomentioning

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%

“…The present work is concerned with the many other combinatorial objects which possess invariants with properties reminiscent of these, such as matroid perspectives and their Las Vergnas polynomial [43] or delta-matroids and their Bollobás-Riordan polynomial [7], which are both matroidal frameworks for certain topological embeddings of graphs in surfaces. Another example, which served as our initial motivation, is arithmetic matroids, introduced by D'Adderio and the third author [20], whose arithmetic Tutte polynomial satisfies a convolution formula (proved by Backman-Lenz [3] in a special case, and then in the present paper in greater generality). To formulate a "Tutte-like" deletion-contraction recurrence for a class of combinatorial objects, we require that each object have an underlying set, and that there are two ways to create new objects by removing elements of this set, deletion and contraction, subject to some axioms.…”

Section: Introductionmentioning

confidence: 96%

See 3 more Smart Citations

Universal Tutte characters via combinatorial coalgebras

Dupont¹,

Fink²,

Moci³

2018

Algebraic Combinatorics

Self Cite

View full text Add to dashboard Cite

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...

show abstract

“…If R is a domain then this means that these two quantities are non-zero. If this condition is satisfied then equations (17), (18) and (19) are equivalent to (20) u…”

Section: Comparison Tomentioning

confidence: 99%

“…We note that equation (20) amounts to saying that N 1 and N 2 are norms. This is another argument for the naturality of our use of norms in our deletion-contraction recurrence formula.…”

Section: Comparison Tomentioning

confidence: 99%

Section: Arithmetic Matroidsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

See 2 more Smart Citations

Universal Tutte characters via combinatorial coalgebras

Dupont¹,

Fink²,

Moci³

2018

Algebraic Combinatorics

Self Cite

View full text Add to dashboard Cite

show abstract

“…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%

The integer cohomology algebra of toric arrangements

Callegaro

Delucchi

2017

Advances in Mathematics

View full text Add to dashboard Cite

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.

show abstract

Matroids Over a Ring

Fink

Moci

2015

Springer INdAM Series

View full text Add to dashboard Cite

Arithmetic matroids, the Tutte polynomial and toric arrangements

Cited by 52 publications

References 37 publications

Universal Tutte characters via combinatorial coalgebras

Universal Tutte characters via combinatorial coalgebras

The integer cohomology algebra of toric arrangements

Matroids Over a Ring

Contact Info

Product

Resources

About