1,720,978 research outputs found
Wonderful Models for Toric Arrangements
No full textWe build a wonderful model for toric arrangements. We develop the "toric analog" of the combinatorics of nested sets, which allows us to define a family of smooth open sets covering the model. In this way, we prove that the model is smooth, and we give a precise geometric and combinatorial description of the normal crossing divisor. © 2011 The Author(s). Published by Oxford University Press. All rights reserved
On the cohomology of arrangements of subtori
Full text linkGiven an arrangement of subtori of arbitrary codimension in a complex torus, we compute the cohomology groups of the complement. Then, by using the Leray spectral sequence, we describe the multiplicative structure on the associated graded cohomology. We also provide a differential model for the cohomology ring, by considering a toric wonderful model and its Morgan algebra. Finally, we focus on the divisorial case, proving a new presentation for the cohomology of toric arrangements
A TUTTE POLYNOMIAL FOR TORIC ARRANGEMENTS
No full textWe introduce a multiplicity Tutte polynomial M(x, y), with applications to zonotopes and toric arrangements. We prove that M(x, y) satisfies a deletion-restriction recursion and has positive coefficients. The characteristic polynomial and the Poincaré polynomial of a toric arrangement are shown to be specializations of the associated polynomial M(x, y), likewise the corresponding polynomials for a hyperplane arrangement are specializations of the ordinary Tutte polynomial. Furthermore, M(1, y) is the Hilbert series of the related discrete Dahmen-Micchelli space, while M(x, 1) computes the volume and the number of integer points of the associated zonotope. © 2011 American Mathematical Society
THE MULTIVARIATE ARITHMETIC TUTTE POLYNOMIAL
Full text linkWe introduce an arithmetic version of the multivariate Tutte polynomial and a quasi-polynomial that interpolates between the two. A generalized Fortuin-Kasteleyn representation with applications to arithmetic colorings and flows is obtained. We give a new and more general proof of the positivity of the coefficients of the arithmetic Tutte polynomial and (in the representable case) a geometrical interpretation of them
The multivariate arithmetic Tutte polynomial
Full text linkWe introduce an arithmetic version of the multivariate Tutte polynomial recently studied by Sokal, and a quasi-polynomial that interpolates between the two. We provide a generalized Fortuin-Kasteleyn representation for representable arithmetic matroids, with applications to arithmetic colorings and flows. We give a new proof of the positivity of the coefficients of the arithmetic Tutte polynomial in the more general framework of pseudo-arithmetic matroids. In the case of a representable arithmetic matroid, we provide a geometric interpretation of the coefficients of the arithmetic Tutte polynomial. © 2012 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France
Polyhedra and parameter spaces for matroids over valuation rings
Full text linkIn this paper we address two of the major foundational questions in the theory of matroids over rings. First, we provide a cryptomorphic axiomatisation, by introducing an analogue of the base polytope for matroids. Second, we describe a parameter space for matroids over a valuation ring, which turns out to be a tropical version of the Bott-Samelson varieties for the full flag variety. Thus a matroid over a valuation ring is a sequence of flags of tropical linear spaces a.k.a. valuated matroids
Arithmetic matroids and Tutte polynomials
Full text linkWe introduce the notion of arithmetic matroid, whose main example is provided by a list of elements in a finitely generated abelian group. We study the representability of its dual, and, guided by the geometry of toric arrangements, we give a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's formula
Universal Tutte characters via combinatorial coalgebras
Full text linkThe Tutte polynomial is the most general invariant of matroids and graphs that can be computed recursively by deleting and contracting edges. We generalize this invariant to any class 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, and produce some new convolution formulae. Our principal tools are combinatorial coalgebras and their convolution algebras. Our results generalize in an intrinsic way the recent results of Krajewski--Moffatt--Tanasa
On a conjecture of Hivert and Thiery about Steenrod operators
No full textWe prove some results related to a conjecture of Hivert and Thiéry about the dimension of the space of q-harmonics. In the process we compute the actions of the involved operators on symmetric and alternating functions, which have some independent interest. We then use these computations to prove other results related to the same conjecture.We prove some results related to a conjecture of Hivert and Thiéry about the dimension of the space of q-harmonics (F. Hivert and N. Thiéry, 2004 [HT]). In the process we compute the actions of the involved operators on symmetric and alternating functions, which have some independent interest. We then use these computations to prove other results related to the same conjecture. © 2012 Elsevier Inc
Combinatorics and topology of toric arrangements defined by root systems.
No full textGiven the toric (or toral) arrangement defined by a root system Φ, we describe the poset of its layers (connected components of intersections) and we count its elements. Indeed we show how to reduce to zero-dimensional layers, and in this case we provide an explicit formula involving the maximal subdiagrams of the affine Dynkin diagram of Φ. Then we compute the Euler characteristic and the Poincare' polynomial of the complement of the arrangement, which is the set of regular points of the torus
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