We explore a combinatorial theory of linear dependency in complex space, complex matroids, with foundations analogous to those for oriented matroids. We give multiple equivalent axiomatizations of complex matroids, showing that this theory captures properties of linear dependency, orthogonality, and determinants over C in much the same way that oriented matroids capture the same properties over R. In addition, our complex matroids come with a canonical S 1 action analogous to the action of C * on a complex vector space.Our phirotopes (analogues of determinants) are the same as those studied previously by Below, Krummeck, and Delucchi [7].We further show that complex matroids cannot have vector axioms analogous to those for oriented matroids.
We initiate the study of group actions on (possibly infinite) semimatroids and geometric semilattices. To every such action is naturally associated an orbit-counting function, a two-variable "Tutte" polynomial and a poset which, in the realizable case, coincides with the poset of connected components of intersections of the associated toric arrangement.In this structural framework we recover and strongly generalize many enumerative results about arithmetic matroids, arithmetic Tutte polynomials and toric arrangements by finding new combinatorial interpretations beyond the realizable case. In particular, we thus find the first class of natural examples of nonrealizable arithmetic matroids. Moreover, under additional conditions these actions give rise to a matroid over Z. As a stepping stone toward our results we also prove an extension of the cryptomorphism between semimatroids and geometric semilattices to the infinite case.
Abstract. Motivated by the work of Salvetti and Settepanella ([24, Remark 4.5]) we introduce certain total orderings of the faces of any shellable regular CW-complex (called shelling-type orderings) that can be used to explicitly construct maximum acyclic matchings of the poset of cells of the given complex. Building on an application of this method to the classical zonotope shellings (i.e., those arising from linear extensions of the tope poset) we describe a class of maximum acyclic matchings for the Salvetti complex of a linear complexified arrangement. To do this, we introduce and study a new purely combinatorial stratification of the Salvetti complex. For the obtained acyclic matchings we give an explicit description of the critical cells that depends only on the chosen linear extension of the poset of regions. It is always possible to choose the linear extension so that the critical cells can be explicitly constructed from the chambers of the arrangement via the bijection to no-broken-circuit sets defined by Jewell and Orlik [17]. Our method generalizes naturally to abstract oriented matroids.
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.
Brenti and Welker have shown that for any simplicial complex X, the face vectors of successive barycentric subdivisions of X have roots which converge to fixed values depending only on the dimension of X. We improve and generalize this result here. We begin with an alternative proof based on geometric intuition. We then prove an interesting symmetry of these roots about the real number -2. This symmetry can be seen via a nice algebraic realization of barycentric subdivision as a simple map on formal power series in two variables. Finally, we use this algebraic machinery with some geometric motivation to generalize the combinatorial statements to arbitrary subdivision methods: any subdivision method will exhibit similar limit behavior and symmetry. Our techniques allow us to compute explicit formulas for the values of the limit roots in the case of barycentric subdivision.Comment: 13 pages, final version, appears in Discrete Mathematics 201
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