Broken Circuit Complexes Research Articles

Lagrangian geometry of matroids

We introduce the conormal fan of a matroid M \operatorname {M} , which is a Lagrangian analog of the Bergman fan of M \operatorname {M} . We use the conormal fan to give a Lagrangian interpretation of the Chern–Schwartz–MacPherson cycle of M \operatorname {M} . This allows us to express the h h -vector of the broken circuit complex of M \operatorname {M} in terms of the intersection theory of the conormal fan of M \operatorname {M} . We also develop general tools for tropical Hodge theory to prove that the conormal fan satisfies Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann relations. The Lagrangian interpretation of the Chern–Schwartz–MacPherson cycle of M \operatorname {M} , when combined with the Hodge–Riemann relations for the conormal fan of M \operatorname {M} , implies Brylawski’s and Dawson’s conjectures that the h h -vectors of the broken circuit complex and the independence complex of M \operatorname {M} are log-concave sequences.

Graded Linearity of Stanley–Reisner Ring of Broken Circuit Complexes

This paper introduces two new notions of graded linear resolution and graded linear quotients, which generalize the concepts of linear resolution property and linear quotient for modules over the polynomial ring A = k[x1, …, xn]. Besides, we compare graded linearity with componentwise linearity in general. For modules minimally generated by a regular sequence in a maximal ideal of A, we show that graded linear quotients imply graded linear resolution property for the colon ideals. On the other hand, we provide specific characterizations of graded linear resolution property for the Stanley–Reisner ring of broken circuit complexes and generalize the results of Van Le and Römer, related to the decomposition of matroids into the direct sum of uniform matroids. Specifically, we show that the matroid can be stratified such that each stratum has a decomposition into uniform matroids. We also present analogs of our results for the Orlik–Terao ideal of hyperplane arrangements which are translations of the corresponding results on matroids.

Finiteness theorems for matroid complexes with prescribed topology

There are finitely many simplicial complexes (up to isomorphism) with a given number of vertices. Translating this fact to the language of h-vectors, there are finitely many simplicial complexes of bounded dimension with h1=k for any natural number k. In this paper we study the question at the other end of the h-vector: Are there only finitely many (d−1)-dimensional simplicial complexes with hd=k for any given k? The answer is no if we consider general complexes, but we focus on three cases coming from matroids: (i) independence complexes, (ii) broken circuit complexes, and (iii) order complexes of geometric lattices. Surprisingly, the answer is yes in all three cases.

Quasi-matroidal classes of ordered simplicial complexes

We introduce the notion of a quasi-matroidal class of ordered simplicial complexes: an approximation to the idea of a matroid cryptomorphism in the landscape of ordered simplicial complexes. A quasi-matroidal class contains pure shifted simplicial complexes and ordered matroid independence complexes. The essential property is that if a fixed simplicial complex belongs to this class for every ordering of its vertex set, then it is a matroid independence complex. Some examples of such classes appear implicitly in the matroid theory literature. We introduce various such classes that highlight different apsects of matroid theory and its similarities with the theory of shifted simplicial complexes. For example, we lift the study of objects like the Tutte polynomial and broken circuit complexes to a quasi-matroidal class that allows us to define such objects for shifted complexes. Furthermore, some of the quasi-matroidal classes are amenable to inductive techniques that can't be applied directly in the context of matroid theory. As an example, we provide a suitable setting to reformulate and extend a conjecture of Stanley about h-vectors of matroids which is expected to be tractable with techniques that are out of reach for matroids alone. This new conjecture holds for pure shifted simplicial complexes and matroids of rank up to 4.

A Gröbner basis for the graph of the reciprocal plane

Given the complement of a hyperplane arrangement, let Γ be the closure of the graph of the map inverting each of its defining linear forms. The characteristic polynomial manifests itself in the Hilbert series of Γ in two different-seeming ways, one due to Terao and the other to Huh and Katz. We define an extension of the no broken circuit complex of a matroid and use it to give a direct Grobner basis argument that the polynomials extracted from the Hilbert series in these two ways agree.

Semi-inverted linear spaces and an analogue of the broken circuit complex

The image of a linear space under inversion of some coordinates is an affine variety whose structure is governed by an underlying hyperplane arrangement. In this paper, we generalize work by Proudfoot and Speyer to show that circuit polynomials form a universal Groebner basis for the ideal of polynomials vanishing on this variety. The proof relies on degenerations to the Stanley-Reisner ideal of a simplicial complex determined by the underlying matroid. If the linear space is real, then the semi-inverted linear space is also an example of a hyperbolic variety, meaning that all of its intersection points with a large family of linear spaces are real.

Flawlessness ofh-vectors of broken circuit complexes

One of the major open questions in matroid theory asks whether the $h$-vector $(h_0,h_1,\ldots,h_s)$ of the broken circuit complex of a matroid $M$ satisfies the following inequalities: $$ h_0\leq h_1\leq \cdots\leq h_{\lfloor s/2\rfloor} \quad \text{and}\qua h_i\le h_{s-i}\ \text{ for }\ 0\leq i \leq \lfloor s/2\rfloor. $$ This paper affirmatively answers the question for matroids that are representable over a field of characteristic zero.

Broken circuit complexes of series–parallel networks

Let (h0,h1,…,hs) with hs≠0 be the h-vector of the broken circuit complex of a series–parallel network M. Let G be a graph whose cycle matroid is M. We give a formula for the difference hs−1−h1 in terms of an ear decomposition of G. A number of applications of this formula are provided, including several bounds for hs−1−h1, a characterization of outerplanar graphs, and a solution to a conjecture on A-graphs posed by Fenton. We also prove that hs−2≥h2 when s≥4.

H-Vectors of matroids and logarithmic concavity

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.

On the Gorensteinness of broken circuit complexes and Orlik–Terao ideals

It is proved that the broken circuit complex of an ordered matroid is Gorenstein if and only if it is a complete intersection. Several characterizations for a matroid that admits such an order are then given, with particular interest in the h-vector of broken circuit complexes of the matroid. As an application, we prove that the Orlik–Terao algebra of a hyperplane arrangement is Gorenstein if and only if it is a complete intersection. Interestingly, our result shows that the complete intersection property (and hence the Gorensteinness as well) of the Orlik–Terao algebra can be determined from the last two nonzero entries of its h-vector.

A GENERALIZATION OF THE SWARTZ EQUALITY

AbstractFor a given (d−1)-dimensional simplicial complex Γ, we denote its h-vector by h(Γ)=(h0(Γ),h1(Γ),. . .,hd(Γ)) and set h−1(Γ)=0. The known Swartz equality implies that if Δ is a (d−1)-dimensional Buchsbaum simplicial complex over a field, then for every 0 ≤ i ≤ d, the inequality ihi(Δ)+(d−i+1)hi−1(Δ) ≥ 0 holds true. In this paper, by using these inequalities, we give a simple proof for a result of Terai (N. Terai, On h-vectors of Buchsbaum Stanley–Reisner rings, Hokkaido Math. J. 25(1) (1996), 137–148) on the h-vectors of Buchsbaum simplicial complexes. We then generalize the Swartz equality (E. Swartz, Lower bounds for h-vectors of k-CM, independence, and broken circuit complexes, SIAM J. Discrete Math. 18(3) (2004/05), 647–661), which in turn leads to a generalization of the above-mentioned inequalities for Cohen–Macaulay simplicial complexes in co-dimension t.

Broken circuit complexes and hyperplane arrangements

We study Stanley-Reisner ideals of broken circuits complexes and characterize those ones admitting a linear resolution or being complete intersections. These results will then be used to characterize arrangements whose Orlik-Terao ideal has the same properties. As an application, we improve a result of Wilf on upper bounds for the coefficients of the chromatic polynomial of a maximal planar graph. We also show that for an ordered matroid with disjoint minimal broken circuits, the supersolvability of the matroid is equivalent to the Koszulness of its Orlik-Solomon algebra.

On the Broken-Circuit Complex of Graphs

It is well known that the Stanley–Reisner ring of a matroid complex is level; this is an algebraic property between the Cohen–Macaulay and Gorenstein properties. A similar result for broken-circuit complexes is no longer true, even for graphs. We show that the Stanley–Reisner ring of the broken-circuit complex, of the cone of any simple graph, is level.

Monomial bases for broken circuit complexes

Let F be a field and let G be a finite graph with a total ordering on its edge set. Richard Stanley noted that the Stanley–Reisner ring F ( G ) of the broken circuit complex of G is Cohen–Macaulay. Jason Brown gave an explicit description of a homogeneous system of parameters for F ( G ) in terms of fundamental cocircuits in G . So F ( G ) modulo this hsop is a finite dimensional vector space. We conjecture an explicit monomial basis for this vector space in terms of the circuits of G and prove that the conjecture is true for two infinite families of graphs. We also explore an application of these ideas to bounding the number of acyclic orientations of G from above.

On the NBC-complexes and [formula omitted]-invariants of abstract convex geometries

An (abstract) convex geometry is a combinatorial abstraction of convexity which is a Moore family with the closure operator satisfying the anti-exchange property. A number of results of matroids on the NBC-complexes (or broken circuit complexes) happen to have some exact analogues in convex geometries: for instance, the Whitney-Rota’s formula of the characteristic function of a matroid, Brylawski’s decomposition of the NBC-complexes, etc. A β -invariant of a convex geometry is derived from the characteristic function in the same way as that of a matroid. We introduce a merging of two convex geometries, which is called a 1-sum, and exhibit the resultant value of the β -invariant of a 1-sum.

NBC Complexes of Convex Geometries

We introduce a notion of a $\textit{broken circuit}$ and an $\textit{NBC complex}$ for an (abstract) convex geometry. Based on these definitions, we shall show the analogues of the Whitney-Rota's formula and Brylawski's decomposition theorem for broken circuit complexes on matroids for convex geometries. We also present an Orlik-Solomon type algebra on a convex geometry, and show the NBC generating theorem.

A new two-variable generalization of the chromatic polynomial

We present a two-variable polynomial, which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of a graph. This new polynomial satisfies both an edge decomposition formula and a vertex decomposition formula. We establish two general expressions for this new polynomial: one in terms of the broken circuit complex and one in terms of the lattice of forbidden colorings. We show that the new polynomial may be considered as a specialization of Stanley's chromatic symmetric function. We finally give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that it can be computed in polynomial time for trees and graphs of restricted pathwidth.

Reversible shellings and an inequality for h-vectors

Several important simplicial complexes including matroid complexes and broken circuit complexes are known to be shellable. We show that the lexicographic order of the bases of a matroid can be reversed to obtain a shelling. We prove that the h-vectors of such reversibly shellable complexes of rank d, which have an empty boundary must satisfy the inequality h o + h 1 … + h i ⩽ h d + h d−1 + … + h d− i for i⩽[ d/2]. In particular, this gives a necessary condition for the h-vector of matroids without coloops.

Sign-Coherent Identities for Characteristic Polynomials of Matroids

We derive an explicit formula for the difference χ(G;λ) − χ(G|X;λ)χ(Tx(G); λ)/(λ − 1), where χ(G;λ) is the characteristic polynomial of a simple matroid G, G|X is the restriction of G to a flat X in G, and Tx(G) is the complete principal truncation of G at the flat X. Two counting proofs of this formula are given. The first uses the critical problem and the second uses the broken-circuit complex. We also derive several inequalities involving Whitney numbers of the first kind and other numerical invariants.

Matroid Shellability, β-Systems, and Affine Hyperplane Arrangements

The broken-circuit complex is fundamental to the shellability and homology of matroids, geometric lattices, and linear hyperplane arrangements. This paper introduces and studies the β-system of a matroid, βnbc(M), whose cardinality is Crapo's β-invariant. In studying the shellability and homology of base-pointed matroids, geometric semilattices, and afflne hyperplane arrangements, it is found that the β-system acts as the afflne counterpart to the broken-circuit complex. In particular, it is shown that the β-system indexes the homology facets for the lexicographic shelling of the reduced broken-circuit complex \(\overline {BC} (M)\), and the basic cycles are explicitly constructed. Similarly, an EL-shelling for the geometric semilattice associated with M is produced,_and it is shown that the β-system labels its decreasing chains.Basic cycles can be carried over from\(\overline {BC} (M)\) The intersection poset of any (real or complex) afflnehyperplane arrangement Α is a geometric semilattice. Thus the construction yields a set of basic cycles, indexed by βnbc(M), for the union ⋃Α of such an arrangement.