There is a one-to-one correspondence between geometric lattices and the intersection lattices of arrangements of homotopy spheres. When the arrangements are essential and fully partitioned, Zaslavsky's enumeration of the cells of the arrangement still holds. An application of the theory shows that all minimal cellular resolutions of matroid Steiner ideals are bounded subcomplexes of homotopy sphere arrangements of the given matroid. As a result the Betti numbers of the ideal are computed and seen to be equivalent to Stanley's formula in the special case of face ideals of independence complexes of 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.
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.
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…
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…
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…
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.
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…
Las Vergnas introduced several lattice structures on the bases of an ordered matroid M by using their external and internal activities. He also noted that when computing the Moebius function of these lattices, it was often zero, although he had no explanation for that fact. The purpose of this paper is to provide a topological reason for this phenomenon. In particular, we show that the order complex of the external lattice L of M is homotopic to the independence complex of the restriction M^*|T where M^* is the dual of M and T is the top element of L. We then compute some examples showing that…
We show that given a finitely generated standard-graded algebra of dimension $d$ over an infinite field, its graded Noether normalizations obey a certain kind of `generic exchange', allowing one to pass between any two of them in at most $d$ steps. We prove analogous generic exchange theorems for minimal reductions of an ideal, minimal complete reductions of a set of ideals, and minimal complete reductions of multigraded $k$-algebras. Finally, we unify all these results into a common axiomatic framework by introducing a new topological-combinatorial structure we call a generic matroid, which i…
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.
Bárány, Kalai, and Meshulam recently obtained a topological Tverberg-type theorem for matroids, which guarantees multiple coincidences for continuous maps from a matroid complex to d-dimensional Euclidean space, if the matroid has sufficiently many disjoint bases. They make a conjecture on the connectivity of k-fold deleted joins of a matroid with many disjoint bases, which would yield a much tighter result - but we provide a counterexample already for the case of k=2, where a tight Tverberg-type theorem would be a topological Radon theorem for matroids. Nevertheless, we prove the topological …
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.
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.
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.
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.
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…
Let K be a field, let R=K[x_1,..., x_m] be a polynomial ring with the standard Z^m-grading (multigrading), let L be a Noetherian multigraded R-module, and let F: E --> G be a finite free multigraded presentation of L over R. Given a choice S of a multihomogeneous basis of E, we construct an explicit canonical finite free multigraded resolution T(F, S) of the R-module L. In the case of monomial ideals our construction recovers the Taylor resolution. A main ingredient of our work is a new linear algebra construction of independent interest, which produces from a representation f over K of a matr…
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.
We introduce the concept of orientation for Lagrangian matroids represented in the flag variety of maximal isotropic subspaces of dimension N in the real vector space of dimension 2N+1. The paper continues the study started in math.CO/0209100.
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…
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…
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.
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.
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.
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…
We introduce motivic zeta functions for matroids. These zeta functions are defined as sums over the lattice points of Bergman fans, and in the realizable case, they coincide with the motivic Igusa zeta functions of hyperplane arrangements. We show that these motivic zeta functions satisfy a functional equation arising from matroid Poincaré duality in the sense of Adiprasito-Huh-Katz. In the process, we obtain a formula for the Hilbert series of the cohomology ring of a matroid, in the sense of Feichtner-Yuzvinsky. We then show that our motivic zeta functions specialize to the topological zeta …
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…
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…
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…
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.
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.
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.
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.
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.
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…
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…
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…
A categorification for the characteristic polynomial of matroids
Institute for Chemical Reaction Design and Discovery, Hokkaido University（北海道大学化学反应设计与发现研究所）；Department of Applied Mathematics, Faculty of Science, Fukuoka University（福冈大学理学部应用数学系）
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…
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.
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…
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 …
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 …
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…
Let $M=(E,\mathcal B)$ be an $\mathbb F_q$-linear matroid; denote by ${\mathcal B}$ the family of its bases, $s(M;α)=\sum_{B\in\mathcal B}\prod_{e \in B} α_e$, where ${α_e\in \mathbb F_q}$. According to the Kontsevich conjecture stated in 1997, the number of nonzero values of $s(M;α)$ is a polynomial with respect to $q$ for all matroids. This conjecture was disproved by P. Brosnan and P. Belkale. In this paper we express the characteristic polynomial of the dual matroid $M^\perp$ in terms of the "correct" Kontsevich formula (for $\mathbb F_q$-linear matroids). This representation generalizes t…
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…
