Rade T. Zivaljevic
L. Lovasz has shown that Sperner's combinatorial lemma admits a generalization involving a matroid defined on the set of vertices of the associated triangulation. Inspired by this result we prove that classical Ky Fan's theorem admits an oriented matroid generalization of similar nature. Ky Fan's theorem is obtained as a corollary if the underlying oriented matroid is chosen to be the alternating matroid C^{m,r} .
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We point out a flaw in papers by Daniel Biss devoted to homotopy type of matroid Grassmannians.
The Bergman fan of a matroid is the intersection of tropical hyperplanes defined by the circuits. A tropical basis is a subset of the circuits set that defines the Bergman fan. Yu and Yuster posed a question whether every simple regular matroid has a unique minimal tropical basis of its Bergman fan, and verified it for graphic, cographic matroids and $R_{10}$. We show every simple binary matroid has a unique minimal tropical basis. Since the regular matroid is binary, we positively answered the question.
The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot and Wakefield, whose properties need to be further explored. In this paper we prove that the Kazhdan-Lusztig polynomials of fan matroids coincide with Motzkin polynomials, which was recently conjectured by Gedeon. As a byproduct, we determine the Kazhdan-Lusztig polynomials of graphic matroids of squares of paths. We further obtain explicit formulas of the Kazhdan-Lusztig polynomials of wheel matroids and whirl matroids. We prove the real-rootedness of the Kazhdan-Lusztig polynomials of these matroids, which provide…
We introduce the notion of real phase structure on rational polyhedral fans in Euclidean space. Such a structure consists of an assignment of affine spaces over $\mathbb{Z}/2\mathbb{Z}$ to each top dimensional face of the fan subject to two conditions. Given an oriented matroid we can construct a real phase structure on the fan of the underlying matroid. Conversely, we show that from a real phase structure on a matroid fan we can produce an orientation of the underlying matroid. Thus real phase structures are cryptomorphic to matroid orientations. The topes of the orientated matroid are recove…
Modular flats of oriented matroids and poset quasi-fibrations
本文研究有向拟阵模平坦的组合学，证明到余秩一模平坦的局部化Salvetti复形的自然映射是偏序集拟纤维化，从而将超可解超平面排列的K(π,1)结果推广到超可解有向拟阵，并构造了许多不可实现的超可解有向拟阵。
If S is a set of matroids, then the matroid M is S-fragile if, for every element e in E(M), either M\e or M/e has no minor isomorphic to a member of S. Excluded-minor characterizations often depend, implicitly or explicitly, on understanding classes of fragile matroids. In certain cases, when F is a minor-closed class of S-fragile matroids, and N is in F, the only members of F that contain N as a minor are obtained from N by increasing the length of fans. We prove that if this is the case, then we can certify it with a finite case-analysis. The analysis involves examining matroids that are at …
Inverse Kazhdan--Lusztig polynomials of fan matroids
本文研究扇形 matroid 的逆 Kazhdan--Lusztig 多项式和逆 Z 多项式，推导了生成函数并得出显式公式，证明了逆 Kazhdan--Lusztig 多项式系数序列的对数凹性。
We construct an intersection product on tropical cycles contained in the Bergman fan of a matroid. To do this we first establish a connection between the operations of deletion and restriction in matroid theory and tropical modifications as defined by Mikhalkin. This product generalises the product of Allermann and Rau, and Allermann and also provides an alternative procedure for intersecting cycles which is not based on intersecting with Cartier divisors. Also, we simplify the definition in the case of one dimensional fan cycles in two dimensional matroidal fans and given an application of th…
Given a matroid $M$ one can define its Orlik-Solomon algebra $OS(M)$ and the Bergman fan $Σ_0(M)$. On the other hand to any rational polyhedral fan $Σ$ one can associate its tropical homology and cohomology groups $\F_\bullet(Σ)$, $\F^\bullet (Σ)$. We will show that the projective Orlik-Solomon algebra $OS_0(M)$ is canonically isomorphic to $\F^\bullet (Σ_0(M))$.
Matroids and semigraphoids are discrete structures abstracting and generalizing linear independence among vectors and conditional independence among random variables, respectively. Despite the different nature of conditional independence from linear independence, deep connections between these two areas are found and still undergoing active research. In this paper, we give a characterization of the embedding of matroids into conditional independence structures and its oriented counterpart, which lead to new axiom systems of matroids and oriented matroids.
The separation theorem of Kirchberger can be proven using a combination of Farkas' Lemma and Caratheodory's Theorem. Since those theorems are at the heart of oriented matroids, we are interested in a generalization of Kirchberger's Theorem to them. This has already been done for rank 3 oriented matroids. Here we prove it for complexes of oriented matroids, which are a generalization of oriented matroids.
The Topological Representation Theorem for (oriented) matroids states that every (oriented) matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a homotopy sphere. In this paper, we use a construction of Engström to show that structure-preserving maps between matroids induce topological mappings between their representations; a result previously known only in the oriented case. Specifically, we show that weak maps induce continuous maps and that the process is a functor from the category of matroids with weak maps to the homotopy category…
Finitary affine oriented matroids
本文开创仿射定向拟阵的公理化研究，建立子式、重定向等基本概念，证明有限情形下的可剥壳性，并利用群作用与多重Tutte多项式推广环面排列的计数性质。
Tropical oriented matroids were defined by Ardila and Develin in 2007 in analogy to (classical) oriented matroids. In this paper we present one tropical analogue for the Topological Representation Theorem.
In this paper we extend the theory of oriented matroids to Lagrangian orthogonal matroids and their representations, and give a completely natural transformation from a representation of a classical oriented matroid to a representation of the same oriented matroid considered as a Lagrangian orthogonal matroid. Classical concepts of orientation and this extension may both be thought of as stratifications of thin Schubert cells into unions of connected components.
In 1967 Herbert Scarf suggested a new proof of Brouwer fixed point theorem based on a surprising analogue of Sperner's lemma. This analogue was motivated by Scarf's work in game theory and mathematical economics. Moreover, Scarf proved a much general version of Sperner's lemma dealing with colorings by vectors. The present paper begins by revisiting Scarf's ideas from the point of view of the basic theory of simplicial cochains in the spirit of author's papers arXiv:1909.00940 and arXiv:2012.13104. After this we get to the main new results of the paper, namely, to a generalization of Scarf res…
We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid determines a subdivision of a product of two simplices, and conjecture that this correspondence is a bijection.
We study the fan structure of Dressians $\Dr(d,n)$ and local Dressians $\Dr(\cM)$ for a given matroid $\cM$. In particular we show that the fan structure on $\Dr(\cM)$ given by the three term Plücker relations coincides with the structure as a subfan of the secondary fan of the matroid polytope $P(\cM)$. As a corollary, we have that a matroid subdivision is determined by its 3-dimensional skeleton. We also prove that the Dressian of the sum of two matroids is isomorphic to the product of the Dressians of the matroids. Finally we focus on indecomposable matroids. We show that binary matroids ar…
Oriented matroids (often called order types) are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a key role in combinatorics, computational geometry, and optimization. This paper surveys prior work and presents an update on the search for bounds on the diameter of the cocircuit graph of an oriented matroid. We review the diameter problem and show the diameter bounds of general oriented matroids reduce to those of uniform oriented matroids. We give the la…
We characterize which systems of sign vectors are the cocircuits of an oriented matroid in terms of the cocircuit graph.
We introduce a new axiomatization of matroid theory that requires the elimination property only among modular pairs of circuits, and we present a cryptomorphic phrasing thereof in terms of Crapo's axioms for flats. This new point of view leads to a corresponding strengthening of the circuit axioms for oriented matroids.
Tropical oriented matroids were defined by Ardila and Develin in 2007. They are a tropical analogue of classical oriented matroids in the sense that they encode the properties of the types of points in an arrangement of tropical hyperplanes -- in much the same way as the covectors of (classical) oriented matroids describe the types in arrangements of linear hyperplanes. Ardila and Develin proved that tropical oriented matroids can be represented as mixed subdivisions of dilated simplices. In this paper we show that this correspondence is a bijection. Moreover, a tropical analogue for the Topol…
We generalize the Varchenko matrix of a hyperplane arrangement to oriented matroids. We show that the celebrated determinant formula for the Varchenko matrix, first proved by Varchenko, generalizes to oriented matroids. It follows that the determinant only depends on the matroid underlying the oriented matroid and analogous formulas hold for cones in oriented matroids. We follow a proof strategy for the original Varchenko formula first suggested by Denham and Hanlon. Besides several technical lemmas this strategy also requires a topological result on supertopes which is of independent interest…
The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial complex of nested sets in the lattice of flats. The Bergman complex is triangulated by the nested set complex, and the two complexes coincide if and only if every connected flat remains connected after contracting along any subflat. This sharpens a result of Ardila-Klivans who showed that the Bergman c…
Zaslavsky conjectures that the bounded complex of a simple hyperplane arrangement is homeomorphic to a ball. We prove this conjecture for the more general uniform affine oriented matroids.
Richter-Gebert proved that every non-Euclidean uniform oriented matroid admits a biquadratic final polynomial. We extend this result to the non-uniform case.
In a recent paper, the first author proved the log-concavity of the coefficients of the characteristic polynomial of a matroid realizable over a field of characteristic 0, answering a long-standing conjecture of Read in graph theory. We extend the proof to all realizable matroids, making progress towards a more general conjecture of Rota-Heron-Welsh. Our proof follows from an identification of the coefficients of the reduced characteristic polynomial as answers to particular intersection problems on a toric variety. The log-concavity then follows from an inequality of Hodge type.
The extension space conjecture of oriented matroid theory states that the space of all one-element, non-loop, non-coloop extensions of a realizable oriented matroid of rank $d$ has the homotopy type of a sphere of dimension $d-1$. We disprove this conjecture by showing the existence of a realizable uniform oriented matroid of high rank and corank 3 with disconnected extension space.
Symmetries of geometric structures such as hyperplane arrangements, point configurations and polytopes have been studied extensively for a long time. However, symmetries of oriented matroids, a common combinatorial abstraction of them, are not understood well. In this paper, we aim to obtain a better understanding of symmetries of oriented matroids. First, we put focus on symmetries of matroids, and give a general construction that generates a $3$-dimensional point configuration with a matroidal symmetry that cannot be realized as a geometric symmetry. The construction is based on the observat…
Salvetti complexes for conditional oriented matroids
本文给出实超平面排列复化补空间与有向拟阵Salvetti复形同伦等价的新证明，该证明无需选择、易于可视化，并推广至条件有向拟阵。
A phased matroid is a matroid with additional structure which plays the same role for complex vector arrangements that oriented matroids play for real vector arrangements. The realization space of an oriented (resp., phased) matroid is the space of vector arrangements in $\mathbb R^n$ (resp., $\mathbb C^n$) that correspond to oriented (resp., phased) matroid, modulo a change of coordinates. According to Mnëv's Universality Theorem, the realization spaces of uniform oriented matroids with rank greater than or equal to $3$ can be as complicated as any open semi-algebraic variety. In contrast, un…
We present an algorithm which takes a graph as input and decides in cubic time if the graph is the cocircuit graph of a uniform oriented matroid. In the affirmative case the algorithm returns the set of signed cocircuits of the oriented matroid. This improves an algorithm proposed by Babson, Finschi and Fukuda. Moreover we strengthen a result of Montellano-Ballesteros and Strausz about crabbed connectivity of cocircuit graphs of uniform oriented matroids.
The Lagrangian geometry of matroids was introduced in [ADH20] through the construction of the conormal fan of a matroid M. We used the conormal fan to give a Lagrangian-geometric interpretation of the h-vector of the broken circuit complex of M: its entries are the degrees of the mixed intersections of certain convex piecewise linear functions $γ$ and $δ$ on the conormal fan of M. By showing that the conormal fan satisfies the Hodge-Riemann relations, we proved Brylawski's conjecture that this h-vector is a log-concave sequence. This sequel explores the Lagrangian combinatorics of matroids, fu…
Oriented matroids can serve as a tool of modeling of collective decision-making processes in contradictory problems of pattern recognition. We present a generalization of the committee techniques of pattern recognition to oriented matroids. A tope committee for an oriented matroid is a subset of its maximal covectors such that every positive halfspace contains more than half of the maximal covectors from this subset. For a large subfamily of oriented matroids their committee structure is quite rich; for example, any maximal chains in their tope posets provide one with information sufficient to…
We prove da Silva's 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result and a result of the third author that the positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a closed ball.
The components of K*-vectors associated to a simple oriented matroid M are the numbers of general or special tope committees for M. Using the principle of inclusion-exclusion, we determine how the reorientations of M on one-element subsets of its ground set affect K*-vectors.
The principle of inclusion-exclusion is applied to subsets of maximal covectors contained in halfspaces of a simple oriented matroid and to convex subsets of its ground set for enumerating tope committees.
Milnor fibrations and oriented matroids
该研究利用定向拟阵为复化实排列构造组合模型，得到与米尔诺纤维同伦等价的有限正则CW复形，证明其同伦类型仅依赖定向拟阵结构，且适用于任意定向拟阵。
The construction of the Varchenko matrix for hyperplane arrangements, first introduced by Alexandre Varchenko, extends naturally to oriented matroids. In this paper, we generalize the theorem of Gao and Zhang by proving that the Varchenko matrix of an oriented matroid has a diagonal form if and only if the pseudohyperplane arrangement corresponding to the oriented matroid is in semigeneral position, i.e. it does not contain a degeneracy. Furthermore, we show that the Varchenko matrix of a pseudoline arrangement has a block diagonal form. This also provides an alternative combinatorial proof fo…
Oriented matroid structures on rank 3 root systems
本文证明了秩3仿射根系在满足Weyl群等变且限制为秩2子系统通常拟阵结构的条件下，具有唯一的定向拟阵结构，并将该唯一性结论推广至有限根系及“干净”秩3根系。
Linkage Axioms for Generic Tropical Oriented Matroids
本文提出了一种统一框架，用于阐明通用热带定向母体及相关对象之间的联系，并引入了新的密码同构和性质。
In 1993, Csima and Sawyer proved that in a non-pencil arrangement of n pseudolines, there are at least $\frac{6}{13}n$ simple points of intersection. Since pseudoline arrangements are the topological representations of reorientation classes of oriented matroids of rank $3$, in this paper, we will use this result to prove by induction that an oriented paving matroid of rank $r \ge 3$ on $n$ elements, where $n \geq 5+ r$, has at least $\frac{12}{13(r-1)} \binom{n}{r-2}$ independent hyperplanes, yielding a new necessary condition for a paving matroid to be orientable.
We present statistics on the decompositions (with respect to a distinguished symmetric 2t-cycle) of vertices of the hypercube graph, whose negative parts are covered by two subsets of the ground set {1,...,t} of the corresponding oriented matroid.
Extension-lifting Bijections for Oriented Matroids
本文提出一种双射，将定向矩阵的基与特殊定向联系起来，通过电路和余电路签名描述，并探讨其在定向矩阵编程和三角剖分中的应用。
In an unpublished manuscript of 1992, Johan Karlander has given an axiomatization of affine oriented matroids. A closer examination of the text revealed an invalid construction and an incorrect argument in the proof of his main theorem. This paper provides an alternative argument to fix and slightly simplify the proof of the main theorem.
The active bijection for oriented matroids (and real hyperplane arrangements, and graphs, as particular cases) is introduced and investigated by the authors in a series of papers. Given any oriented matroid defined on a linearly ordered ground set, we exhibit one particularité of its bases, which we call its active basis, with remarkable properties. It preserves activities (for oriented matroids in the sense of Las Vergnas, for matroid bases in the sense of Tutte), as well as some active partitions of the ground set associated with oriented matroids and matroid bases. It yields a canonical bij…
Leclerc and Zelevinsky, motivated by the study of quasi-commuting quantum flag minors, introduced the notions of strongly separated and weakly separated collections. These notions are closely related to the theory of cluster algebras, to the combinatorics of the double Bruhat cells, and to the totally positive Grassmannian. A key feature, called the purity phenomenon, is that every maximal by inclusion strongly (resp., weakly) separated collection of subsets in $[n]$ has the same cardinality. In this paper, we extend these notions and define $\mathcal{M}$-separated collections for any oriented…
The signed Varchenko Determinant for Complexes of Oriented Matroids
本研究将超平面配置的带号Varchenko矩阵推广到有向拟阵复形，证明其行列式具备良好因式分解特性，拓展了超平面配置与有向拟阵领域的既有成果。
In this paper we generalise the even directed cycle problem, which asks whether a given digraph contains a directed cycle of even length, to orientations of regular matroids. We define non-even oriented matroids generalising non-even digraphs, which played a central role in resolving the computational complexity of the even dicycle problem. Then we show that the problem of detecting an even directed circuit in a regular matroid is polynomially equivalent to the recognition of non-even oriented matroids. Our main result is a precise characterisation of the class of non-even oriented bond matroi…
The Las Vergnas' strong map conjecture, states that any strong map of oriented matroids $f:\mathcal{M}_1\rightarrow\mathcal{M}_2$ can be factored into extensions and contractions. The conjecture is known to be false due to a construction by Richter-Gebert, he find a non-factorizable strong map $f:\mathcal{M}_1\rightarrow\mathcal{M}_2$, however in his example $\mathcal{M}_1$ is not realizable. The problem that whether there exists a non-factorizable strong map between realizable oriented matroids still remains open. In this paper we provide a counterexample to the strong map conjecture on reali…
For a symmetric 2t-cycle in the tope graph of a simple oriented matroid M on the ground set {1,...,t}, where t is even, we describe decompositions of topes and subtopes of M with respect to the subtopes corresponding to the edges of the symmetric cycle.
We introduce the active partition of the ground set of an oriented matroid perspective (or quotient, or strong map) on a linearly ordered ground set. The reorientations obtained by arbitrarily reorienting parts of the active partition share the same active partition. This yields an equivalence relation for the set of reorientations of an oriented matroid perspective, whose classes are enumerated by coefficients of the Tutte polynomial, and a remarkable partition of the set of reorientations into boolean lattices, from which we get a short direct proof of a 4-variable expansion formula for the …
Las Vergnas and Hamidoune studied the number of circuits needed to determine an oriented matroid. In this paper we investigate this problem and some new variants, as well as their interpretation in particular classes of matroids. We present general upper and lower bounds in the setting of general connected orientable matroids, leading to the study of subgraphs of the base graph and the intersection graph of circuits. We then consider the problem for uniform matroids which is closely related to the notion of (connected) covering numbers in Design Theory. Finally, we also devote special attentio…
In this paper we present a definition of oriented Lagrangian symplectic matroids and their representations. Classical concepts of orientation and this extension may both be thought of as stratifications of thin Schubert cells into unions of connected components. The definitions are made first in terms of a combinatorial axiomatisation, and then again in terms of elementary geometric properties of the Coxeter matroid polytope. We also generalise the concept of rank and signature of a quadratic form to symplectic Lagrangian matroids in a surprisingly natural way.
We introduce the conormal fan of a matroid M, which is a Lagrangian analog of the Bergman fan of M. We use the conormal fan to give a Lagrangian interpretation of the Chern-Schwartz-MacPherson cycle of M. This allows us to express the h-vector of the broken circuit complex of M in terms of the intersection theory of the conormal fan of 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, when com…
We present statistics on the decompositions (with respect to a distinguished symmetric 2t-cycle) of vertices of the hypercube graph, whose negative parts are regarded as disjoint unions of two subsets of the ground set {1,...,t} of the corresponding oriented matroid.
Neighborly polytopes are those that maximize the number of faces in each dimension among all polytopes with the same number of vertices. Despite their extremal properties they form a surprisingly rich class of polytopes, which has been widely studied and is the subject of many open problems and conjectures. In this paper, we study the enumeration of neighborly polytopes beyond the cases that have been computed so far. To this end, we enumerate neighborly oriented matroids --- a combinatorial abstraction of neighborly polytopes --- of small rank and corank. In particular, if we denote by OM($r,…
On the Possibilities of Defining Infinite Oriented Matroids
本文探讨了通过将有限定向矩阵的公理系统提升到无限设置是否能保持对偶性的可能性，发现两种公理系统均无法保持对偶性。
Tropical Geometry of Rado Matroids
本文通过热带几何方法刻画无环拟阵 Chow 环的单纯生成元乘积，证明热带超平面与 Bergman 扇的稳定交对应 Rado 拟阵对偶的 Bergman 扇，推广了 Backman、Eur 和 Simpson 的结果。
Oriented cobicircular matroids are $GSP$
本文研究定向拟阵的 Hadwiger 猜想，通过构造不具有正余线的严格 gammoid 无限族否定相关猜想，并提出更简单技术证明定向余双圆拟阵等是 $GSP$。
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