The f -vector of a representable-matroid complex is log-concave

Lenz, Matthias

doi:10.1016/j.aam.2013.07.001

Cited by 41 publications

(44 citation statements)

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“…We demonstrate this method by proving the log-concavity of h-vectors of two simplicial complexes associated to a matroid, when the matroid is representable over a field of characteristic zero. Other illustrations can be found in [16,18,19].…”

Section: Introduction and Resultsmentioning

confidence: 98%

h-Vectors of matroids and logarithmic concavity

Huh

2015

Advances in Mathematics

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Let M be a matroid on E, representable over a field of characteristic zero. We show that h-vectors of the following simplicial complexes are log-concave:1. The matroid complex of independent subsets of E. 2. The broken circuit complex of M relative to an ordering of E.The first implies a conjecture of Colbourn on the reliability polynomial of a graph, and the second implies a conjecture of Hoggar on the chromatic polynomial of a graph. The proof is based on the geometric formula for the characteristic polynomial of Denham, Garrousian, and Schulze.

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Section: Introduction and Resultsmentioning

confidence: 98%

h-Vectors of matroids and logarithmic concavity

Huh

2015

Advances in Mathematics

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“…To deduce Corollary 4.4 from the results of [1], one proceeds by showing that the signed f -polynomial f 0 (M )t r − f 1 (M )t r−1 + · · · + (−1) r f r (M ) of the rank r matroid M coincides with the reduced characteristic polynomial of an auxiliary rank r +1 matroid M constructed from M , the so-called free co-extension of M . 11 This identity was originally proved by Brylawski [7] and subsequently rediscovered by Lenz [21]. 4.6.…”

Section: 2mentioning

confidence: 89%

Hodge theory in combinatorics

Baker

2017

Bull. Amer. Math. Soc.

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Abstract. If G is a finite graph, a proper coloring of G is a way to color the vertices of the graph using n colors so that no two vertices connected by an edge have the same color. (The celebrated four-color theorem asserts that if G is planar, then there is at least one proper coloring of G with four colors.) By a classical result of Birkhoff, the number of proper colorings of G with n colors is a polynomial in n, called the chromatic polynomial of G. Read conjectured in 1968 that for any graph G, the sequence of absolute values of coefficients of the chromatic polynomial is unimodal: it goes up, hits a peak, and then goes down. Read's conjecture was proved by June Huh in a 2012 paper making heavy use of methods from algebraic geometry. Huh's result was subsequently refined and generalized by Huh and Katz (also in 2012), again using substantial doses of algebraic geometry. Both papers in fact establish log-concavity of the coefficients, which is stronger than unimodality.The breakthroughs of the Huh and Huh-Katz papers left open the more general Rota-Welsh conjecture, where graphs are generalized to (not necessarily representable) matroids, and the chromatic polynomial of a graph is replaced by the characteristic polynomial of a matroid. The Huh and HuhKatz techniques are not applicable in this level of generality, since there is no underlying algebraic geometry to which to relate the problem. But in 2015 Adiprasito, Huh, and Katz announced a proof of the Rota-Welsh conjecture based on a novel approach motivated by but not making use of any results from algebraic geometry. The authors first prove that the Rota-Welsh conjecture would follow from combinatorial analogues of the hard Lefschetz theorem and Hodge-Riemann relations in algebraic geometry. They then implement an elaborate inductive procedure to prove the combinatorial hard Lefschetz theorem and Hodge-Riemann relations using purely combinatorial arguments.We will survey these developments.

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“…The log-concavity conjecture for f i pMq was proved in [34] by combining a geometric construction of [27] and a matroid-theoretic construction of Brylawski [6]. Given a spanning subset M of a d-dimensional vector space over a field k, one can construct a d-dimensional smooth projective variety XpMq over k and globally generated line bundles L 1 , L 2 on XpMq so that…”

Section: 5mentioning

confidence: 99%

Combinatorial Applications of the Hodge–riemann Relations

Huh

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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Logarithmic concavity is a property of a sequence of real numbers, occurring throughout algebra, geometry, and combinatorics. A sequence of real numbers a 0 , . . . , a d is log-concave ifWhen all the entries are positive, the log-concavity implies unimodality, a property easier to visualize: the sequence is unimodal if there is an index i such thatA rich variety of log-concave and unimodal sequences arising in combinatorics can be found in the surveys [5,55,56]. For an extensive discussion of log-concavity and its applications in probability and statistics, see [12,39,48].Why do natural and interesting sequences often turn out to be log-concave? Below we give one of many possible explanations, from the viewpoint of "standard conjectures". To illustrate, we discuss three combinatorial sequences appearing in [56, Problem 25], in Sections 2.4, 2.5, and 2.8. Another heuristic, based on the physical principle that the entropy of a system should be concave as a function of the energy, can be found in [44].Let X be a mathematical object of "dimension" d. Often it is possible to construct from X in a natural way a graded vector space over the real numbersa symmetric bilinear map P : ApXqˆApXq Ñ R, and a graded linear map L : A ‚ pXq Ñ A ‚`1 pXq that is symmetric with respect to P. The linear operator L usually comes in as a member of a family KpXq, a convex cone in the space of linear operators on ApXq. 1 For example, ApXq may be the cohomology of real pq, qq-forms on a compact Kähler manifold [22], the ring of algebraic cycles modulo homological equivalence on a smooth projective variety [24], McMullen's algebra generated by the Minkowski summands of a simple convex polytope [42], the combinatorial intersection cohomology of a convex polytope [33], the reduced Soergel bimodule of a Coxeter group element [16], or the Chow ring of a matroid defined in Section 2.6.

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The f -vector of a representable-matroid complex is log-concave

Cited by 41 publications

References 3 publications

h-Vectors of matroids and logarithmic concavity

h-Vectors of matroids and logarithmic concavity

Hodge theory in combinatorics

Combinatorial Applications of the Hodge–riemann Relations

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