Lagrangian Matroids: Representations of Type $B_n$

Lagrangian subspaces, delta-matroids and four-term relations

Finite order invariants (Vassiliev invariants) of knots are expressed in terms of weight systems, that is, functions on chord diagrams satisfying the four-term relations. Weight systems have graph analogues, so-called $4$-invariants of graphs, i.e. functions on graphs that satisfy the four-term relations for graphs. Each $4$-invariant determines a weight system. The notion of weight system is naturally generalized for the case of embedded graphs with an arbitrary number of vertices. Such embedded graphs correspond to links; to each component of a link there corresponds a vertex of an embedded …

Lagrangian geometry of matroids

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…

Lagrangian Grassmannians的Baker--Bowler理论

Baker--Bowler theory for Lagrangian Grassmannians

本文将Baker--Bowler理论扩展至Lagrangian Grassmannian，通过简化二次曲面定义了antisymmetric matroids，提供了基于circuits的等价定义，并证明了相关同伦定理与二次关系满足条件，统一并推广了现有理论。

Graphical representations of graphic frame matroids

A frame matroid M is graphic if there is a graph G with cycle matroid isomorphic to M. In general, if there is one such graph, there will be many. Zaslavsky has shown that frame matroids are precisely those having a representation as a biased graph; this class includes graphic matroids, bicircular matroids, and Dowling geometries. Whitney characterized which graphs have isomorphic cycle matroids, and Matthews characterised which graphs have isomorphic graphic bicircular matroids. In this paper, we give a characterization of which biased graphs give rise to isomorphic graphic frame matroids.

A Topological Representation Theorem for Tropical Oriented Matroids: Part I

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.

格路径拟阵的高效表示

Efficient Representation of Lattice Path Matroids

本文基于分层秘密共享方案、分层拟阵端口刻画及多项式规模隔离权函数，提出了在有限域上高效构造格路径拟阵表示的确定性算法。

关于拟阵的扩张性质与表示的一个注记

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。

Representation of matroids with a modular plane

We prove that if M is a vertically 4-connected matroid with a modular flat X of rank at least three, then every representation of M | X over a finite field F extends to a unique F-representation of M. A corollary is that when F has order q, any vertically 4-connected matroid with a PG(2, F)-restriction is either F-representable or has a U_{2, q^2+1}-minor. We also show that no excluded minor for the class of F-representable matroids has a PG(2, F)-restriction.

A Topological Representation Theorem for Tropical Oriented Matroids: Part II

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…

A two-dimensional topological representation theorem for matroid polytopes of rank 4

The Folkman-Lawrence topological representation theorem, which states that every (loop-free) oriented matroid of rank $r$ can be represented as a pseudosphere arrangement on the $(r-1)$-dimensional sphere $S^{r-1}$, is one of the most outstanding results in oriented matroid theory. In this paper, we provide a lower-dimensional version of the topological representation theorem for uniform matroid polytopes of rank $4$. We introduce $2$-weak configurations of points and pseudocircles ($2$-weak PPC configurations) on $S^2$ and prove that every uniform matroid polytope of rank $4$ can be represent…

Homotopy sphere representations for matroids

For any rank $r$ oriented matroid $M$, a construction is given of a "topological representation" of $M$ by an arrangement of homotopy spheres in a simplicial complex which is homotopy equivalent to $S^{r-1}$. The construction is completely explicit and depends only on a choice of maximal flag in $M$. If $M$ is orientable, then all Folkman-Lawrence representations of all orientations of $M$ embed in this representation in a homotopically nice way.

Recognizing signed-graphic matroids: Cylinder flips and the importance of column scaling

In this paper, we investigate the importance of column scaling in relating two signed-graphic representations of the same matroid. We used the Sage Mathematics software to generate many examples of signed-graphic matroids and their signed-graphic representations. Our examples show that column scaling is sometimes necessary in order to transform one signed-graphic representation into another; moreover, there exist many collections of signed-graphic representations that row-reduce to the same standard form. We also discuss an interesting matroid-preserving operation on a signed graph, which we c…

Representations of torsion-free arithmetic matroids

We study the representability problem for torsion-free arithmetic matroids. By using a new operation called "reduction" and a "signed Hermite normal form", we provide and implement an algorithm to compute all the representations, up to equivalence. As an application, we disprove two conjectures about the poset of layers and the independence poset of a toric arrangement.

拟阵的Chow环作为置换表示

Chow Rings of Matroids as Permutation Representations

研究拟阵自同构群在Chow环上的诱导作用，证明其为置换作用，并将庞加莱对偶与硬莱夫谢茨定理提升至该作用，提出相关猜想。

An Algorithmic Proof of the Piff--Welsh Theorem on Transversal Matroid Representations

A fundamental theorem of matroid theory establishes that a transversal matroid is representable over fields of any characteristic. It was proved in 1970 by Piff and Welsh: their proof is elegant and concise and, moveover, constructive. However it is far from being algorithmic, in terms of suggesting a step-by-step procedure for deriving a collection of vectors over a given base field representing the transversal matroid induced by a given set system. In this note we recast Piff and Welsh's proof in algorithmic form.

Representations of bicircular lift matroids

Bicircular lift matroids are a class of matroids defined on the edge set of a graph. For a given graph $G$, the circuits of its bicircular lift matroid are the edge sets of those subgraphs of $G$ that contain at least two cycles, and are minimal with respect to this property. The main result of this paper is a characterization of when two graphs give rise to the same bicircular lift matoid, which answers a question proposed by Irene Pivotto. In particular, aside from some appropriately defined "small" graphs, two graphs have the same bicircular lift matroid if and only if they are $2$-isomorph…

Representing some non-representable matroids

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…

Inequivalent Representations of Matroids over Prime Fields

It is proved that for each prime field $GF(p)$, there is an integer $f(p)$ such that a 4-connected matroid has at most $f(p)$ inequivalent representations over $GF(p)$. We also prove a stronger theorem that obtains the same conclusion for matroids satisfying a connectivity condition, intermediate between 3-connectivity and 4-connectivity that we term "$k$-coherence". We obtain a variety of other results on inequivalent representations including the following curious one. For a prime power $q$, let ${\mathcal R}(q)$ denote the set of matroids representable over all fields with at least $q$ elem…

Matroids of Gain Graphs in Applied Discrete Geometry

A G-gain graph is a graph whose oriented edges are labeled invertibly from a group G. Zaslavsky proposed two matroids of G-gain graphs, called frame matroids and lift matroids, and investigated linear representations of them. Each matroid has a canonical representation over a field F if G is isomorphic to a subgroup of F^{\times} in the case of frame matroids or G is isomorphic to an additive subgroup of F in the case of lift matroids. The canonical representation of the frame matroid of a complete graph is also known as a Dowling geometry, as it was first introduced by Dowling for finite grou…

Almost balanced biased graph representations of frame matroids

Given a 3-connected biased graph $Ω$ with a balancing vertex, and with frame matroid $F(Ω)$ nongraphic and 3-connected, we determine all biased graphs $Ω'$ with $F(Ω') = F(Ω)$. As a consequence, we show that if $M$ is a 4-connected nongraphic frame matroid represented by a biased graph $Ω$ having a balancing vertex, then $Ω$ essentially uniquely represents $M$. More precisely, all biased graphs representing $M$ are obtained from $Ω$ by replacing a subset of the edges incident to its unique balancing vertex with unbalanced loops.

Matrix Representations of Transversal Matroids

A transversal matroid $M$ of rank $r$ on $[n]$ can be associated to a family of binary matrices corresponding to different presentations of $M$. We describe those matrices which arise from unique maximal presentations of size $r$- giving a complete characterization when $r=2$-, give an algorithm for producing such a matrix, and show that determining whether a matrix corresponds to a maximal presentation is related to determining the tropical rank of arbitrary matrices.

Matroids, hereditary collections and simplicial complexes having boolean representations

Inspired by the work of Izakhian and Rhodes, a theory of representation of hereditary collections by boolean matrices is developed. This corresponds to representation by finite $\vee$-generated lattices. The lattice of flats, defined for hereditary collections, lattices and matrices, plays a central role in the theory. The representations constitute a lattice and the minimal and strictly join irreducible elements are studied, as well as various closure operators.

Foundations of matroids I: Matroids without large uniform minors

The foundation of a matroid is a canonical algebraic invariant which classifies representations of the matroid up to rescaling equivalence. Foundations of matroids are pastures, a simultaneous generalization of partial fields and hyperfields. Using deep results due to Tutte, Dress-Wenzel, and Gelfand-Rybnikov-Stone, we give a presentation for the foundation of a matroid in terms of generators and relations. The generators are certain "cross-ratios" generalizing the cross-ratio of four points on a projective line, and the relations encode dependencies between cross-ratios in certain low-rank co…

Lift theorems for representations of matroids over pastures

Pastures are a class of field-like algebraic objects which include both partial fields hyperfields and have nice categorical properties. We prove several lift theorems for representations of matroids over pastures, including a generalization of Pendavingh and van Zwam's Lift Theorem for partial fields. By embedding the earlier theory into a more general framework, we are able to establish new results even in the case of lifts of partial fields, for example the conjecture of Pendavingh--van Zwam that their lift construction is idempotent. We give numerous applications to matroid representations…

Face numbers of Engström representations of matroids

A classic problem in matroid theory is to find subspace arrangements, specifically hyperplane and pseudosphere arrangements, whose intersection posets are isomorphic to a prescribed geometric lattice. Engström recently showed how to construct an infinite family of such subspace arrangements, indexed by the set of finite regular CW complexes. In this note, we compute the face numbers of these representations (in terms of the face numbers of the indexing complexes) and give upper bounds on the total number of faces in these objects. In particular, we show that, for a fixed rank, the total number…

On the representability of the bi-uniform matroid

Every bi-uniform matroid is representable over all sufficiently large fields. But it is not known exactly over which finite fields they are representable, and the existence of efficient methods to find a representation for every given bi-uniform matroid has not been proved. The interest of these problems is due to their implications to secret sharing. The existence of efficient methods to find representations for all bi-uniform matroids is proved here for the first time. The previously known efficient constructions apply only to a particular class of bi-uniform matroids, while the known genera…

通过 pasture 态射表示拟阵

Representing matroids via pasture morphisms

基于 pasture 框架与拟阵基础，提出计算拟阵基础及 pasture 间所有态射的算法，高效解决拟阵可定向性、不可表示性及有限域上所有表示等表示问题。

均匀超图的Chow环和增强Chow环及其q-类比

Chow rings and augmented Chow rings of uniform matroids and their $q$-analogs

本文研究了均匀超图和q-均匀超图的Chow环和增强Chow环的表示及其q-类比，推导了Hilbert级数和Frobenius级数的公式，并揭示了Charney--Davis量的非负性及组合解释。

Matrix representations of frame and lifted-graphic matroids correspond to gain functions

Let $M$ be a 3-connected matroid and let $\mathbb F$ be a field. Let $A$ be a matrix over $\mathbb F$ representing $M$ and let $(G,\mathcal B)$ be a biased graph representing $M$. We characterize the relationship between $A$ and $(G,\mathcal B)$, settling four conjectures of Zaslavsky. We show that for each matrix representation $A$ and each biased graph representation $(G,\mathcal{B})$ of $M$, $A$ is projectively equivalent to a canonical matrix representation arising from $G$ as a gain graph over $\mathbb F^+$ or $\mathbb F^\times$ realizing $\mathcal{B}$. Further, we show that the projectiv…

Tableaux in the Whitney Module of a Matroid

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 …

Fair representation in the intersection of two matroids

For a simplicial complex ${\mathcal C}$ denote by $β({\mathcal C})$ the minimal number of edges from ${\mathcal C}$ needed to cover the ground set. If ${\mathcal C}$ is a matroid then for every partition $A_1, \ldots, A_m$ of the ground set there exists a set $S \in {\mathcal C}$ meeting each $A_i$ in at least $\frac{|A_i|}{β({\mathcal C})}$ elements. We conjecture that a slightly weaker result is true for the intersections of two matroids: if ${\mathcal D}={\mathcal P} \cap {\mathcal Q}$, where ${\mathcal P},{\mathcal Q}$ are matroids on the same ground set $V$ and $β({\mathcal P}), β({\mathc…

折叠代数拟阵：特征刚性与几乎熵分离

Folded-Algebraic Matroids: Characteristic Rigidity and Almost-Entropic Separation

该研究引入折叠代数拟阵，证明其与几乎熵拟阵为真包含关系，构造出几乎熵但非折叠代数的拟阵，还得到具有不相容特征要求的连通端口，明确了相关表示类的分离区域。

Lifts of matroid representations over partial fields

There exist several theorems which state that when a matroid is representable over distinct fields F_1,...,F_k, it is also representable over other fields. We prove a theorem, the Lift Theorem, that implies many of these results. First, parts of Whittle's characterization of representations of ternary matroids follow from our theorem. Second, we prove the following theorem by Vertigan: if a matroid is representable over both GF(4) and GF(5), then it is representable over the real numbers by a matrix such that the absolute value of the determinant of every nonsingular square submatrix is a powe…

$q$-matroids 的表示性 via 排序度码

Representability of $q$-matroids via rank-metric codes

本文研究了 $q$-matroids 的多线性表示性，通过排序度码的自然可除性条件定义了 $m$-多线性表示性，并证明了非平凡均匀 $q$-matroids 无法纯多线性表示，且非帕普斯 $q$-matroid 的块大小至少为 9。

二元矩阵的表示张量积

Represented Tensor Products of Binary Matroids

本文分类了二元矩阵表示张量积的正则性、余图性、图性及二元性，指出非自由因子乘积的正则性条件，并证明固定因子时性质不依赖表示域。

Representation of matroids

In this paper we give a necessary and sufficient criterion for representability of a matroid over an algebraic closed field. This leads to an algorithm, based on an extension of Groebner Bases, in order to decide if a given matroid is representable over such a field.

Algebraic matroids and Frobenius flocks

We show that each algebraic representation of a matroid $M$ in positive characteristic determines a matroid valuation of $M$, which we have named the {\em Lindström valuation}. If this valuation is trivial, then a linear representation of $M$ in characteristic $p$ can be derived from the algebraic representation. Thus, so-called rigid matroids, which only admit trivial valuations, are algebraic in positive characteristic $p$ if and only if they are linear in characteristic $p$. To construct the Lindström valuation, we introduce new matroid representations called flocks, and show that each alge…

Topological representations of matroid maps

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…

双圈拟阵类只有有限个排除子式

There are only a finite number of excluded minors for the class of bicircular matroids

本文证明了双圈拟阵类仅有有限个排除子式，核心方法是利用偏置图和拟图拟阵表示，证明秩至少为十的排除子式必为拟图拟阵并包含已知小排除子式，从而得出结论。

A Topological Representation Theorem for Oriented Matroids

We present a new direct proof of a topological representation theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Sch{ö}nflies theorem. As an application, we show that one can read off oriented matroids from arrangements of embedded spheres of codimension one, even if wild spheres are involved.

Entropic matroids and their representation

This paper investigates entropic matroids, that is, matroids whose rank function is given as the Shannon entropy of random variables. In particular, we consider $p$-entropic matroids, for which the random variables each have support of cardinality $p$. We draw connections between such entropic matroids and secret-sharing matroids and show that entropic matroids are linear matroids when $p = 2,3$ but not when $p = 9$. Our results leave open the possibility for $p$-entropic matroids to be linear whenever $p$ is prime, with particular cases proved here. Applications of entropic matroids to coding…

Unlabeled equivalence for matroids representable over finite fields

We present a new type of equivalence for representable matroids that uses the automorphisms of the underlying matroid. Two $r\times n$ matrices $A$ and $A'$ representing the same matroid $M$ over a field $F$ are {\it geometrically equivalent representations} of $M$ if one can be obtained from the other by elementary row operations, column scaling, and column permutations. Using geometric equivalence, we give a method for exhaustively generating non-isomorphic matroids representable over a finite field $GF(q)$, where $q$ is a power of a prime.

Representations of weakly multiplicative arithmetic matroids are unique

An arithmetic matroid is weakly multiplicative if the multiplicity of at least one of its bases is equal to the product of the multiplicities of its elements. We show that if such an arithmetic matroid can be represented by an integer matrix, then this matrix is uniquely determined. This implies that the integer cohomology ring of a centred toric arrangement whose arithmetic matroid is weakly multiplicative is determined by its poset of layers. This partially answers a question asked by Callegaro-Delucchi.

Topological representation of matroids from diagrams of spaces

Swartz proved that any matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a sphere. This was an unexpected extension from the oriented matroid case, but unfortunately the construction is not explicit. Anderson later provided an explicit construction, but had to use cell complexes of high dimensions that are homotopy equivalent to lower dimensional spheres. Using diagrams of spaces we give an explicit construction of arrangements in the right dimensions. Swartz asked if it is possible to arrange spheres of codimension two, and we provide…

Common Information, Matroid Representation, and Secret Sharing for Matroid Ports

Linear information and rank inequalities as, for instance, Ingleton inequality, are useful tools in information theory and matroid theory. Even though many such inequalities have been found, it seems that most of them remain undiscovered. Improved results have been obtained in recent works by using the properties from which they are derived instead of the inequalities themselves. We apply here this strategy to the classification of matroids according to their representations and to the search for bounds on secret sharing for matroid ports.

New Representations of Matroids and Generalizations

We extend the notion of matroid representations by matrices over fields and consider new representations of matroids by matrices over finite semirings, more precisely over the boolean and the superboolean semirings. This idea of representations is generalized naturally to include also hereditary collections. We show that a matroid that can be directly decomposed as matroids, each of which is representable over a field, has a boolean representation, and more generally that any arbitrary hereditary collection is superboolean-representable.

旗空间、拟阵舒尔代数与斯坦伯格表示

Flag Space, Matroidal Schur Algebras and the Steinberg Representation

研究通过生成元和关系定义拟阵舒尔代数的扩展版本，在射影几何中关联其与\(GL(n,q)\)模斯坦伯格表示结构，一般情形下证明其有三角或里迪分解，还得到原始代数新描述并阐明与旗空间联系。

低秩的拟阵表示

Matroidal representations of low rank

本文研究有限群布尔正则表示的热带子表示（等价于具特定自同构的拟阵），完全分类了秩3时的此类子表示，阿贝尔群情形推广了戈隆尺，还引入了由有限集等价关系得到的拟阵类。

Boolean Representations of Matroids and Lattices

We introduce a new representation concept for lattices by boolean matrices, and utilize it to prove that any matroid is boolean representable. We show that such a representation can be easily extracted from a representation of the associated lattice of flats of the matroid, leading also to a tighter bound on the representation's size. Consequently, we obtain a linkage of boolean representations with geometry in a very natural way.

Representability of Matroids by c-Arrangements is Undecidable

For a natural number $c$, a $c$-arrangement is an arrangement of dimension $c$ subspaces satisfying the following condition: the sum of any subset of the subspaces has dimension a multiple of $c$. Matroids arising as normalized rank functions of $c$-arrangements are also known as multilinear matroids. We prove that it is algorithmically undecidable whether there exists a $c$ such that a given matroid has a $c$-arrangement representation, or equivalently whether the matroid is multilinear. It follows that certain network coding problems are also undecidable. In the proof, we introduce a general…

Representations of automorphism groups on the homology of matroids

Given a group $G$ of automorphisms of a matroid $M$, we describe the representations of $G$ on the homology of the independence complex of the dual matroid $M^*$. These representations are related with the homology of the lattice of flats of $M$, and (when $M$ is realizable) with the top cohomology of a hyperplane arrangement. Finally we analyze in detail the case of the complete graph, which has applications to algebraic geometry.

Matroid Horn functions

Hypergraph Horn functions were introduced as a subclass of Horn functions that can be represented by a collection of circular implication rules. These functions possess distinguished structural and computational properties. In particular, their characterizations in terms of implicate-duality and the closure operator provide extensions of matroid duality and the Mac Lane-Steinitz exchange property of matroid closure, respectively. In the present paper, we introduce a subclass of hypergraph Horn functions that we call matroid Horn functions. We provide multiple characterizations of matroid Horn …

泰特着色与拟阵特征多项式的$\alpha$-表示

The $α$-representation for the Tait coloring and for the characteristic polynomial of matroid

该研究提出正则拟阵特征多项式的$\alpha$-表示，推广了此前的图流多项式公式，且该技术可用于表示三次双连通平面图的泰特着色数。

Parallel Translates of Represented Matroids

Given an $\Bbb{F}$-represented matroid $(M,ρ)$ with the ground set $[m]$, the representation $ρ$ naturally defines a hyperplane arrangement $\mathcal{A}_ρ$. We will study its parallel translates $\mathcal{A}_{ρ,{g}}$ of $\mathcal{A}_ρ$ for all ${ g}\in \mathbb{F}^m$. Its intersection semi-lattices $L(\mathcal{A}_{ρ,{ g}})$ and the characteristic polynomials $χ(\mathcal{A}_{ρ,{ g}},t)$ will be classified by the intersection lattice of the derived arrangement $\mathcal{A}_{δρ}$, which is a hyperplane arrangement associated with the derived matroid $(δM,δρ)$ and also known as the discriminantal a…