Dedicated to Denis Higgs. 25 pages, 3 figures. Submitted for publication in the Journal of Combinatorial Theory (A). See arXiv:math.CO/0409028 arXiv:math.CO/0409080 for preparatory work on this subject
We study the combinatorial, algebraic and geometric properties of the free product operation on matroids. After giving cryptomorphic definitions of free product in terms of independent sets, bases, circuits, closure, flats and rank function, we show that free product, which is a noncommutative operation, is associative and respects matroid duality. The free product of matroids $M$ and $N$ is maximal with respect to the weak order among matroids having $M$ as a submatroid, with complementary contraction equal to $N$. Any minor of the free product of $M$ and $N$ is a free product of a repeated truncation of the corresponding minor of $M$ with a repeated Higgs lift of the corresponding minor of $N$. We characterize, in terms of their cyclic flats, matroids that are irreducible with respect to free product, and prove that the factorization of a matroid into a free product of irreducibles is unique up to isomorphism. We use these results to determine, for K a field of characteristic zero, the structure of the minor coalgebra $\cal C$ of a family of matroids $\cal M$ that is closed under formation of minors and free products: namely, $\cal C$ is cofree, cogenerated by the set of irreducible matroids belonging to $\cal M$.
We study the problem of factorization for residues of generalized biadjoint scalar scattering amplitudes $m^{(k)}_n$, introduced by Cachazo, Early, Guevara and Mizera (CEGM), involving multi-dimensional residues which factorize generically into $k$-ary products of lower-point generalized biadjoint amplitudes of the same type $m^{(k)}_{n_1}\cdots m^{(k)}_{n_k}$, where $n_1+\cdots +n_k = n+k(k-1)$, noting that smaller numbers of factors arise as special cases. Such behavior is governed geometrically by regular matroid subdivisions of hypersimplices and cones in the positive tropical Grassmannian…
The expansion axiom of matroids requires only the existence of some kind of independent sets, not the uniqueness of them. This causes that the base families of some matroids can be reduced while the unions of the base families of these matroids remain unchanged. In this paper, we define unique expansion matroids in which the expansion axiom has some extent uniqueness; we define union minimal matroids in which the base families have some extent minimality. Some properties of them and the relationship between them are studied. First, we propose the concepts of secondary base and forming base fam…
A conjecture of Graver from 1991 states that the generic $3$-dimensional rigidity matroid is the unique maximal abstract $3$-rigidity matroid with respect to the weak order on matroids. Based on a close similarity between the generic $d$-dimensional rigidity matroid and the generic $C_{d-2}^{d-1}$-cofactor matroid from approximation theory, Whiteley made an analogous conjecture in 1996 that the generic $C_{d-2}^{d-1}$-cofactor matroid is the unique maximal abstract $d$-rigidity matroid for all $d\geq 2$. We verify the case $d=3$ of Whiteley's conjecture in this paper. A key step in our proof i…
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 notion of thin sums matroids was invented to extend the notion of representability to non-finitary matroids. A matroid is tame if every circuit-cocircuit intersection is finite. We prove that a tame matroid is a thin sums matroid over a finite field k if and only if all its finite minors are representable over k. We expect that the method we use to prove this will make it possible to lift many theorems about finite matroids representable over a finite field to theorems about tame thin sums matroids over these fields. We give three examples of this: various characterisations of binary tame …
We prove that the maximum size of a simple binary matroid of rank $r \geq 5$ with no AG(3,2)-minor is $\binom{r+1}{2}$ and characterise those matroids achieving this bound. When $r \geq 6$, the graphic matroid $M(K_{r+1})$ is the unique matroid meeting the bound, but there are a handful of smaller examples. In addition, we determine the size function for non-regular simple binary matroids with no AG(3,2)-minor and characterise the matroids of maximum size for each rank.
Generalizing a well known theorem for finite matroids, we prove that for every (infinite) connected matroid M there is a unique tree T such that the nodes of T correspond to minors of M that are either 3-connected or circuits or cocircuits, and the edges of T correspond to certain nested 2-separations of M. These decompositions are invariant under duality.
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.
Holt and Klee have recently shown that every (generic) LP orientation of the graph of a $d$-polytope satisfies a directed version of the $d$-connectivity property, i.e. there are $d$ internally disjoint directed paths from a unique source to a unique sink. We introduce two new classes HK and HK* of oriented matroids (OMs) by enforcing this property and its dual interpretation in terms of line shellings, respectively. Both classes contain all representable OMs by the Holt-Klee theorem. While we give a construction of an infinite family of non-HK* OMs, it is not clear whether there exists any no…
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…
$q$-Matroids are defined on complemented modular support lattices. Minors of length 2 are of four types as in a "classical" matroid. Tutte polynomials $τ(x,y)$ of matroids are calculated either by recursion over deletion/contraction of single elements, by an enumeration of bases with respect to internal/external activities, or by substitution $x \to (x-1),\; y \to (y-1)$ in their rank generating functions $ρ(x,y)$. The $q$-analogue of the passage from a Tutte polynomial to its corresponding RGF is straight-forward, but the analogue of the reverse process $x \to (x-1),\; y \to (y-1)$ is more de…
We show that additive induced-hereditary properties of coloured hypergraphs can be uniquely factorised into irreducible factors. Our constructions and proofs are so general that they can be used for arbitrary concrete categories of combinatorial objects; we provide some examples of such combinatorial objects.
The prism graph is the dual of the complete graph on five vertices with an edge deleted, $K_5\backslash e$. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac's infinite families of graphs and four infinite families of non-regular matroids determined by Oxley, there are only three possibilities for a matroid in this class: it is isomorphic to the dual of the generalized parallel connectio…
We introduce and study filtrations of a matroid on a linearly ordered ground set, which are particular sequences of nested sets. A given basis can be decomposed into a uniquely defined sequence of bases of minors, such that these bases have an internal/external activity equal to 1/0 or 0/1 (in the sense of Tutte polynomial activities). This decomposition, which we call the active filtration/partition of the basis, refines the known partition of the ground set into internal and external elements with respect to a given basis. It can be built by a certain closure operator, which we call the acti…
In this paper, we investigate three problems concerning the toric ideal associated to a matroid. Firstly, we list all matroids $\mathcal M$ such that its corresponding toric ideal $I_{\mathcal M}$ is a complete intersection. Secondly, we handle with the problem of detecting minors of a matroid $\mathcal M$ from a minimal set of binomial generators of $I_{\mathcal M}$. In particular, given a minimal set of binomial generators of $I_{\mathcal M}$ we provide a necessary condition for $\mathcal M$ to have a minor isomorphic to $\mathcal U_{d,2d}$ for $d \geq 2$. This condition is proved to be suff…
For an integer $n>2$, a rank-$n$ matroid is called an $n$-spike if it consists of $n$ three-point lines through a common point such that, for all $k\in\{1, 2, ..., n - 1\}$, the union of every set of $k$ of these lines has rank $k+1$. Spikes are very special and important in matroid theory. In 2003 Wu found the exact numbers of $n$-spikes over fields with 2, 3, 4, 5, 7 elements, and the asymptotic values for larger finite fields. In this paper, we prove that, for each prime number $p$, a $GF(p$) representable $n$-spike $M$ is only representable on fields with characteristic $p$ provided that $…
We consider integrable, category O-modules of indecomposable symmetrizable Kac-Moody algebras. We prove that unique factorization of tensor products of irreducible modules holds in this category, upto twisting by one dimensional modules. This generalizes a fundamental theorem of Rajan for finite dimensional simple Lie algebras over C. Our proof is new even for the finite dimensional case, and uses an interplay of representation theory and combinatorics to analyze the Kac-Weyl character formula.
Interaction between skew-representability, tensor products, extension properties, and rank inequalities
ELTE Eötvös Loránd University（罗兰大学）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）；London School of Economics and Political Science（伦敦政治经济学院）
It is well-known that all finite connected graphs have a unique prime factor decomposition (PFD) with respect to the strong graph product which can be computed in polynomial time. Essential for the PFD computation is the construction of the so-called Cartesian skeleton of the graphs under investigation. In this contribution, we show that every connected thin hypergraph H has a unique prime factorization with respect to the normal and strong (hypergraph) product. Both products coincide with the usual strong graph product whenever H is a graph. We introduce the notion of the Cartesian skeleton o…
Non-unique factorizations theory, which started in algebraic number theory, over the years has expanded into several areas of mathematics. Here, we propose yet another branching. We show that some concepts of factorizations theory, such as half factorial and weak half factorial properties can be translated via Cayley graphs into graph theory. It is proved, that subset S of abelian group G is half factorial, if and only if the Cayley digraph Cay(G; S) is geodetical, e.g., simple paths connecting a fixed pair of vertices have the same length. Further, it is shown that the voltage digraph natural…
A finite Coxeter group $W$ has a natural metric $d$ and if $\mathcal{M}$ is a subset of $W$, then for each $u\in W$, there is $q\in \mathcal{M}$ such that $d(u,q)=d(u,\mathcal{M})$. Such $q$ is not unique in general but if $\mathcal{M}$ is a Coxeter matroid, then it is unique, and we define a retraction $\mathcal{R}^m_{\mathcal{M}}\colon W\to \mathcal{M}\subset W$ so that $\mathcal{R}^m_{\mathcal{M}}(u)=q$. The $T$-fixed point set $Y^T$ of a $T$-orbit closure $Y$ in a flag variety $G/B$ is a Coxeter matroid, where $G$ is a semisimple algebraic group, $B$ is a Borel subgroup, and $T$ is a maxim…
