Boris Shapiro, Arkady Vaintrob
In this short note we make a few remarks on a class of generalized incidence matrices whose matroids do not depend on the orientation of the underlying graph and natural commutative algebras associated to such matrices.
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A new class of bi-transversal matroids
针对双横截拟阵的构造问题，本文为无向图G及其任意顶点子集W关联了一类新的双横截拟阵，并给出计算其基数的显式公式。
In his pioneering paper on matroids in 1935, Whitney obtained a characterization for binary matroids and left a comment at end of the paper that the problem of characterizing graphic matroids is the same as that of characterizing matroids which correspond to matrices (mod 2) with exactly two ones in each column. Later on Tutte obtained a characterization of graphic matroids in terms of forbidden minors in 1959. It is clear that Whitney indicated about incidence matrices of simple undirected graphs. Here we introduce the concept of a segment binary matroid which corresponds to matrices over $\m…
A mixed graph is a graph with some directed edges and some undirected edges. We introduce the notion of mixed matroids as a generalization of mixed graphs. A mixed matroid can be viewed as an oriented matroid in which the signs over a fixed subset of the ground set have been forgotten. We extend to mixed matroids standard definitions from oriented matroids, establish basic properties, and study questions regarding the reorientations of the unsigned elements. In particular we address in the context of mixed matroids the P-connectivity and P-orientability issues which have been recently introduc…
Connectoids I: a universal end space theory
本文引入connectoid概念以刻画离散对象连通结构，构建通用端空间理论统一并拓展多类结构的端空间，证明方向与端一一对应且正规树可表示其端。
Let $G$ be a finite undirected multigraph with no self-loops. The Jacobian $\operatorname{Jac}(G)$ is a finite abelian group associated with $G$ whose cardinality is equal to the number of spanning trees of $G$. There are only a finite number of biconnected graphs $G$ such that the exponent of $\operatorname{Jac}(G)$ equals $2$ or $3$. The definition of a Jacobian can also be extended to regular matroids as a generalization of graphs. We prove that there are finitely many connected regular matroids $M$ such that $\operatorname{Jac}(M)$ has exponent $2$ and characterize all such matroids.
In this paper, we give a cancellation-free antipode formula (for uniform matroids) for the restriction-contraction matroid Hopf algebra, using the technique of splitting and merging via a sign-reversing involution. The cancellation-free formula expresses the antipode of uniform matroids as a sum over certain ordered set partitions.
Gordon introduced a class of matroids $M(n)$, for prime $n\ge 2$, such that $M(n)$ is algebraically representable, but only in characteristic $n$. Lindström proved that $M(n)$ for general $n\ge 2$ is not algebraically representable if $n>2$ is an even number, and he conjectured that if $n$ is a composite number it is not algebraically representable. We introduce a new kind of matroid called {\it harmonic matroids}, of which full algebraic matroids are an example. We prove the conjecture in this more general case.
In this note we introduce a sufficient condition for the Orlik-Solomon algebra associated to a matroid M to be l-adic and we prove that this condition is necessary when M is binary (in particular graphic). Moreover, this result cannot be extended to the class of all matroids.
We introduce the matroid-minor coalgebra C, which has labeled matroids as distinguished basis, and coproduct given by splitting a matroid into a submatroid and complementary contraction all possible ways. We introduce two new bases for C; the first of these is is related to the distinguished basis by Mobius inversion over the rank-preserving weak order on matroids, the second by Mobius inversion over the suborder excluding matroids that are irreducible with respect to the free product operation. We show that the subset of each of these bases corresponding to the set of irreducible matroids is …
This paper is an initial inquiry into the structure of the Hopf algebra of matroids with restriction-contraction coproduct. Using a family of matroids introduced by Crapo in 1965, we show that the subalgebra generated by a single point and a single loop in the dual of this Hopf algebra is free.
In this paper, we give a cancellation-free antipode formula for the matroid-minor Hopf algebra. We then explore applications of this formula. For example, the cancellation-free formula expresses the antipode of uniform matroids as a sum over certain ordered set partitions. We also prove that all matroids over any hyperfield (in the sense of Baker and Bowler) have cancellation-free antipode formulas; furthermore, the cancellations in the antipode are independent of the hyperfield structure and only depend on the underlying matroid.
Matroids arising from algebraic shifting
本文研究了通过代数移位得到的matroid结构，揭示了移位超图与matroid基集的关系，并给出了相关matroid的分类和例子。
A family C of circuits of a matroid M is a linear class if, given a modular pair of circuits in C}, any circuit contained in the union of the pair is also in C. The pair (M,C) can be seen as a matroidal generalization of a biased graph. We introduce and study an Orlik-Solomon type algebra determined by (M,C). If C is the set of all circuits of M this algebra is the Orlik-Solomon algebra of M.
The $OS$ algebra $A$ of a matroid $M$ is a graded algebra related to the Whitney homology of the lattice of flats of $M$. In case $M$ is the underlying matroid of a hyperplane arrangement \A in $\C^r$, $A$ is isomorphic to the cohomology algebra of the complement $\C^r\setminus \bigcup \A.$ Few examples are known of pairs of arrangements with non-isomorphic matroids but isomorphic $OS$ algebras. In all known examples, the Tutte polynomials are identical, and the complements are homotopy equivalent but not homeomorphic. We construct, for any given simple matroid $M_0$, a pair of infinite famili…
The Whitney module of a matroid is a natural analogue of the tensor algebra of the exterior algebra of a vector space that takes into account the dependencies of a matroid. In this paper we indicate the role that tableaux can play in describing the Whitney module. We will use our results to describe a basis of the Whitney module of a certain class of matroids known as freedom matroids (also known as Schubert, or shifted matroids). The doubly multilinear submodule of the Whitney module is a representation of the symmetric group. We will describe a formula for the multiplicity of hook shapes in …
Generalized Weight Polynomials of Codes through Flats and Orlik-Solomon Algebras of Matroids
本文提出多种从平展格、Orlik-Solomon代数及循环平展格计算拟阵广义重量多项式的方法，并给出线性码广义重量谱的应用及示例。
Pattern Recognition on Oriented Matroids: Symmetric Cycles in the Hypercube Graphs. VI
本文探讨二元数组精确模型表示的线性代数、组合与应用层面，并以字符串的两幅线性代数画像加以说明。
This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a) algebraic matroids, we expose cryptomorphisms making them accessible to techniques from commutative algebra. This allows us to introduce for each circuit in an algebraic matroid an invariant called circuit polynomial, generalizing the minimal poly- nomial in classical Galois theory, and studying the matroid structure with multivariate methods. For (b) matroid…
Orlik-Solomon algebras determine the Tutte polynomials of matroids of rank at most three
本文证明了Falk关于秩为3的简单拟阵的Orlik-Solomon代数确定其Tutte多项式的猜想，且该结论对所有秩不超过3的简单拟阵均成立。
Fix a principal ideal domain $k$. In this article we associate to a (weighted) matroid $M$ a quasi-hereditary algebra $R(M)$ defined over $k$ such that matroid duality corresponds to Ringel duality of quasi-hereditary algebras. The representation theory of these algebras is related to work of Schechtman-Varchenko and Brylawski-Varchenko. In characteristic zero, our algebras are also closely related to work of Kook-Reiner-Stanton and Denham.
Tropical matroid Schubert varieties and the graded Möbius algebra
本文引入热带Matroid Schubert变种，证明其热带上同调环与Matroid的分级Möbius代数同构，扩展了Arrangement Schubert变种到任意Matroid。
Recently, M.~Baker and N.~Bowler introduced the notion of matroids over hyperfields as a unifying theory of various generalizations of matroids. In this paper we generalize the notion of minors and direct sums from ordinary matroids to matroids over hyperfields. Using this we generalize the classical construction of matroid-minor Hopf algebras to the case of matroids over hyperfields.
In this paper we consider the original and different generalizations of Postnikov-Shapiro algebra which enumerate forests and trees of graphs, see~\cite{PSh}. Our main result is that the algebra counting forests depends only on graphical matroid and converse. Also we generalize algebras for a hypergraph. For this, we define spanning forests and trees of a hypergraph and the corresponding "hypergraphical" matroid. We present $3$ different equivalent definitions of spanning forests and trees, which can be read independently from other parts of the paper.
Matroid complexes and Orlik-Solomon algebras
本文为超可解拟阵的Orlik-Solomon代数构造组合拟自由微分分次模型，推广Kontsevich的容许图cdga，并据此证明其Koszul性。
Resonance varieties and Lie algebras of matroids and hyperplane arrangements
本文综述拟阵 Orlik-Solomon 代数上的共振簇、完整李代数与完整 Chen 李代数，比较可实现与不可实现拟阵的异同并提出开放问题。
We prove the Lefschetz property for a certain class of finite-dimensional Gorenstein algebras associated to matroids. Our result implies the Sperner property of the vector space lattice. More generally, it is shown that the modular geometric lattice has the Sperner property. We also discuss the Gröbner fan of the defining ideal of our Gorenstein algebra.
We endow the set of isomorphic classes of matroids with a new Hopf algebra structure, in which the coproduct is implemented via the combinatorial operations of restriction and deletion. We also initiate the investigation of dendriform coalgebra structures on matroids and introduce a monomial invariant which satisfy a convolution identity with respect to restriction and deletion.
Matroids are ubiquitous in modern combinatorics. As discovered by Gelfand, Goresky, MacPherson and Serganova there is a beautiful connection between matroid theory and the geometry of Grassmannians: realizable matroids correspond to torus orbits in Grassmannians. Further, as observed by Fink and Speyer general matroids correspond to classes in the $K$-theory of Grassmannians. This yields in particular a geometric description of the Tutte polynomial. In this review we describe all these constructions in detail, and moreover we generalise some of them to polymatroids. More precisely, we study th…
New families of matroids are constructed in this note. These new fami- lies are derived from the concept of linearly independent set family (LISF) introduced by Eicker and Ewald [Linear Algebra and its Applications 388 (2004) 173-191]. The proposed construction generalizes in a natural way the well known class of vectorial matroids over a field.
Cycle polytopes of matroids have been introduced in combinatorial optimization as a generalization of important classes of polyhedral objects like cut polytopes and Eulerian subgraph polytopes associated to graphs. Here we start an algebraic and geometric investigation of these polytopes by studying their toric algebras, called cycle algebras, and their defining ideals. Several matroid operations are considered which determine faces of cycle polytopes that belong again to this class of polyhedral objects. As a key technique used in this paper, we study certain minors of given matroids which yi…
It has recently been shown that infinite matroids can be axiomatized in a way that is very similar to finite matroids and permits duality. This was previously thought impossible, since finitary infinite matroids must have non-finitary duals. In this paper we illustrate the new theory by exhibiting its implications for the cycle and bond matroids of infinite graphs. We also describe their algebraic cycle matroids, those whose circuits are the finite cycles and double rays, and determine their duals. Finally, we give a sufficient condition for a matroid to be representable in a sense adapted to …
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 $\F^\bullet (Σ_0(M))$.
We show that the algebraic rank of divisors on certain graphs is related to the realizability problem of matroids. As a consequence, we produce a series of examples in which the algebraic rank depends on the ground field. We use the theory of metrized complexes to show that equality between the algebraic and combinatorial rank is not a sufficient condition for smoothability of divisors, thus giving a negative answer to a question posed by Caporaso, Melo, and the author.
Using linear algebra methods we study certain algebraic properties of monomial rings and matroids. Let I be a monomial ideal in a polynomial ring over an arbitrary field. If the Rees cone of I is quasi-ideal, we express the normalization of the Rees algebra of I in terms of an Ehrhart ring. We introduce the basis Rees cone of a matroid (or a polymatroid) and study their facets. Some applications to Rees algebras are presented. It is shown that the basis monomial ideal of a matroid (or a polymatroid) is normal.
We introduce an idempotent analogue of the exterior algebra for which the theory of tropical linear spaces (and valuated matroids) can be seen in close analogy with the classical Grassmann algebra formalism for linear spaces. The top wedge power of a tropical linear space is its Plucker vector, which we view as a tensor, and a tropical linear space is recovered from its Plucker vector as the kernel of the corresponding wedge multiplication map. We prove that an arbitrary d-tensor satisfies the tropical Plucker relations (valuated exchange axiom) if and only if the d-th wedge power of the kerne…
Connectoids II: existence of normal trees
本文研究了连接体中正常树的存在性，扩展了无向图中正常树的概念，通过推广Jung的分散集特征，证明了连接体具有正常生成树当且仅当其基础集可以以某种称为可数分离数的方式良序。
A Leray model for the Orlik-Solomon algebra
该研究构造了拟阵上Leray模型的组合推广形式，通过组合爆破数据给出双分次交换微分分次代数的表示，利用Gröbner基构造单项式基，证明其与拟阵的经典Orlik-Solomon代数拟同构。
A Note on Extension Properties and Representations of Matroids
本文利用拟阵与多拟阵的扩张性质作为各类表示存在的必要条件，经计算机辅助在八、九元拟阵中发现新的非线性表示例子，并刻画了含井字棋构型的九点稀疏铺砌拟阵。
A classical problem in Distance Geometry, with multiple practical applications (in molecular structure determination, sensor network localization etc.) is to find the possible placements of the vertices of a graph with given edge lengths. For minimally rigid graphs, the double-exponential Gröbner Bases algorithm with an elimination order can be applied, in theory, but it is impractical even for small instances. By relating the problem to the computation of circuit polynomials in the Cayley-Menger ideal, we recently proposed an algebraic-combinatorial approach and an elimination algorithm for c…
We associate to a sufficiently generic oriented matroid program and choice of linear system of parameters a finite dimensional algebra, whose representation theory is analogous to blocks of Bernstein--Gelfand--Gelfand category $\mathcal O$. When the data above comes from a generic linear program for a hyperplane arrangement, we recover the algebra defined by Braden--Licata--Proudfoot--Webster. Applying our construction to nonlinear oriented matroid programs provides a large new class of algebras. For Euclidean oriented matroid programs, the resulting algebras are quasi-hereditary and Koszul, a…
Graded Betti numbers of graded Möbius algebras of uniform matroids
本文推导了任意均匀拟阵的分次默比乌斯代数定义理想的分次贝蒂数精确公式，并研究了任意拟阵的分次默比乌斯代数具有线性表示的条件。
Matroids in OSCAR
本文介绍了新型计算机代数系统OSCAR中处理拟阵的模块相关内容，聚焦拟阵的实现空间与周环计算，体现拟阵在多数学领域的关联作用。
Let k be a field of characteristic zero. We consider graded subalgebras A of k[x_1,...,x_m]/(x_1^2,...,x_m^2) generated by d linearly independant linear forms. Representations of matroids over k provide a natural description of the structure of these algebras. In return, the numerical properties of the Hilbert function of A yield some information about the Tutte polynomial of the corresponding matroid. Isomorphism classes of these algebras correspond to equivalence classes of hyperplane arrangements under the action of the general linear group.
Exterior algebras in matroid theory
本文为$\mathbb{F}_1^{\pm}$-algebras引入外代数的类似物，为拟阵提供新的密码同态，并展示了从环和幂等半域恢复相应代数的方法。
The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. Hyperbolic polynomials give rise to a class of (hyperbolic) matroids which properly contains the class of matroids representable over the complex numbers. This connection was used by the second author to construct counterexamples to algebraic (stronger) versions of the generalized Lax conjecture by considering a non-representable hyperbolic matroid. The Vámos matroid and a generalization of it are, prior to this work, the only known instances of non-representable…
We extend the notion of representation of a matroid to algebraic structures that we call skew partial fields. Our definition of such representations extends Tutte's definition, using chain groups. We show how such representations behave under duality and minors, we extend Tutte's representability criterion to this new class, and we study the generator matrices of the chain groups. An example shows that the class of matroids representable over a skew partial field properly contains the class of matroids representable over a skew field. Next, we show that every multilinear representation of a ma…
Transfer systems give matroids only for cyclic $p$-groups
本文研究等变代数中的转移系统，证明其最小生成集构成拟阵仅当格为线性序或群为素数幂阶循环群，并计算了该拟阵的不变量。
This paper examines the category $\mathbf{Mat}_{\bullet}$ of pointed matroids and strong maps from the point of view of Hall algebras. We show that $\mathbf{Mat}_{\bullet}$ has the structure of a finitary proto-exact category - a non-additive generalization of exact category due to Dyckerhoff-Kapranov. We define the algebraic K-theory $K_* (\mathbf{Mat}_{\bullet})$ of $\mathbf{Mat}_{\bullet}$ via the Waldhausen construction, and show that it is non-trivial, by exhibiting injections $$π^s_n (\mathbb{S}) \hookrightarrow K_n (\mathbf{Mat}_{\bullet})$$ from the stable homotopy groups of spheres fo…
Associative spectra of graph algebras are examined with the help of homomorphisms of DFS trees. Undirected graphs are classified according to the associative spectra of their graph algebras; there are only three distinct possibilities: constant 1, powers of 2, and Catalan numbers. Associative and antiassociative digraphs are described, and associative spectra are determined for certain families of digraphs, such as paths, cycles, and graphs on two vertices.
We will start from the beginning and define a matroid and its Orlik-Solomon algebra and holonomy Lie algebra, but first we give some background from topology and cohomology. A (central) hyperplane arrangement is a finite number of subspaces of codimension one in a finite dimensional vector space over R, C or even finite fields. The space X which is the complement of the union of the hyperplanes has interesting properties. If you consider R^n the cohomology is however rather trivial, there is just an H^0, since X is a disjoint union of contractible sets. But it is not a trivial task to compute …
Zonotopal algebra is the study of a family of pairs of dual vector spaces of multivariate polynomials that can be associated with a list of vectors X. It connects objects from combinatorics, geometry, and approximation theory. The origin of zonotopal algebra is the pair (D(X),P(X)), where D(X) denotes the Dahmen-Micchelli space that is spanned by the local pieces of the box spline and P(X) is a space spanned by products of linear forms. The first main result of this paper is the construction of a canonical basis for D(X). We show that it is dual to the canonical basis for P(X) that is already …
Failure of the Lefschetz property for the Graphic Matroid
本文针对 Maeno 和 Numata 关于拟阵基生成多项式定义的 Artinian Gorenstein 代数具有强 Lefschetz 性质的猜想，构造了图拟阵反例，并通过构造高阶 Hessian 矩阵核中的非零元素证明其退化性。
Koszul Graded Möbius Algebras and Strongly Chordal Graphs
本文研究了拟阵分次 Möbius 代数何时为 Koszul 代数，基于图的性质刻画了图圈拟阵的 Koszul 分次 Möbius 代数，并通过边排序给出了强弦图的新刻画。
Let A be a (central) arrangement of hyperplanes in a finite dimension complex vector space V. Let M(A) be the dependence matroid determined by A. The Orlik-Solomon algebra OS(M) of a matroid M is the exterior algebra on the points modulo the ideal generated by circuit boundaries. The algebra OS(M) is isomorphic to the cohomology algebra of the complement in V of the union of the hyperplanes of A. The Tutte polynomial T(x,y) of M is a powerful invariant of the matroid M. When M(A) is a rank three matroid and A is the complexification of a real arrangement, we prove that OS(M) determines T(x,y).…
Let us consider a truncated matroid $M_Γ^{r}$ of rank $r$ of a graphic matroid of a graph $Γ$. The basis for $M_Γ^{r}$ is the set of the forests with $r$ edges in $Γ$. We consider this basis generating function and compute its Hessian. In this paper, we show that the Hessian of the basis generating function of the truncated matroid of the graphic matroid of the complete or complete bipartite graph does not vanish by calculating the eigenvalues of the Hessian matrix. Moreover, we show that the Hessian matrix of the basis generating function of the truncated matroid of the graphic matroid of the…
On the Ehrhart Theory of Generalized Symmetric Edge Polytopes
本文通过组合学和格基技术，将SEPs的性质扩展到广义SEPs，并证明广义SEPs不一定是γ非负的，但通过删除两个元素可得到γ非负的SEPs，支持Ohsugi和Tsuchiya的猜想。
A biased graph is a graph with a class of selected circles ("cycles", "circuits"), called "balanced", such that no theta subgraph contains exactly two balanced circles. A biased graph has two natural matroids, the frame matroid and the lift matroid. A classical question in matroid theory is whether a matroid can be embedded in a projective geometry. There is no known general answer, but for matroids of biased graphs it is possible to give algebraic criteria. Zaslavsky has previously given such criteria for embeddability of biased-graphic matroids in Desarguesian projective spaces; in this pape…
Reconstruction of oriented matroids from Varchenko-Gelfand algebras
本文研究特征不等于2的系数环上的Varchenko-Gelfand代数，证明在余维2的一般超平面排列中，有向拟阵可从滤过和分次VG代数恢复，并给出一般情况下的重构算法。
The Orlik-Solomon algebra of a matroid can be considered as a quotient ring over the exterior algebra E. At first we study homological properties of E-modules as e.g. complexity, depth and regularity. In particular, we consider modules with linear injective resolutions. We apply our results to Orlik-Solomon algebras of matroids and give formulas for the complexity, depth and regularity of such rings in terms of invariants of the matroid. Moreover, we characterize those matroids whose Orlik-Solomon ideal has a linear projective resolution and compute in these cases the Betti numbers of the idea…
Explorations of Matroid Complexes
受Kontsevich图复形启发，本文系统研究拟阵复形，通过构造删除-收缩双复形并建立Hopf代数结构，证明广泛acyclicity结果，并通过计算小规模复形检测非平凡同调，提出与奇轮拟阵相关的猜想。
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