相关论文: Matroids Arising From Nested Sequences of Flats In…
The theory of matroids or combinatorial geometries originated in linear algebra and graph theory, and has deep connections with many other areas, including field theory, matching theory, submodular optimization, Lie combinatorics, and total…
This study aims to shed light on new (sub)classes of matroids originating from cluster algebras and investigate their properties. We focus on what we call cluster matroids and build some results on them. Then, we point out a relationship…
We call a class $\mathcal{M}$ of matroids hereditary if it is closed under flats. We denote by $\mathcal{M}^{ext}$ the class of matroids $M$ that is in $\mathcal{M}$, or has an element $e$ such that $M \backslash e$ is in $\mathcal{M}$. We…
A finite unit norm tight frame is a collection of $r$ vectors in $\mathbb{R}^n$ that generalizes the notion of orthonormal bases. The affine finite unit norm tight frame variety is the Zariski closure of the set of finite unit norm tight…
This article is a survey of matroid theory aimed at algebraic geometers. Matroids are combinatorial abstractions of linear subspaces and hyperplane arrangements. Not all matroids come from linear subspaces; those that do are said to be…
Nested matroids were introduced by Crapo in 1965 and have appeared frequently in the literature since then. A flat of a matroid $M$ is Hamiltonian if it has a spanning circuit. A matroid $M$ is nested if and only if its Hamiltonian flats…
In 1961, Dirac showed that chordal graphs are exactly the graphs that can be constructed from complete graphs by a sequence of clique-sums. In an earlier paper, by analogy with Dirac's result, we introduced the class of $GF(q)$-chordal…
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…
Rook matroids were recently introduced by the author and Alexandersson as matroids whose bases arise from certain restricted rook placements on a skew-shaped board. They were shown to be a subclass of transversal matroids and positroids. We…
A graph is chordal if every cycle of length at least four has a chord. In 1961, Dirac characterized chordal graphs as those graphs that can be built from complete graphs by repeated clique-sums. Generalizing this, we consider the class of…
Consider a matroid equipped with a labeling of its ground set to an abelian group. We define the label of a subset of the ground set as the sum of the labels of its elements. We study a collection of problems on finding bases and common…
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…
We characterize the shifted simple graphs and the $3$-uniform shifted hypergraphs whose inverse image under exterior shifting is the set of bases of a matroid: those are exactly the hypergraphs whose hyperedges form an initial lex-segment.…
A matroid is a combinatorial structure that captures and generalizes the algebraic concept of linear independence under a broader and more abstract framework. Matroids are closely related with many other topics in discrete mathematics, such…
Building sets were introduced in the study of wonderful compactifications of hyperplane arrangement complements and were later generalized to finite meet-semilattices. Convex geometries, the duals of antimatroids, offer a robust…
Classical designs and their (projective) q-analogs can both be viewed as designs in matroids, using the matroid of all subsets of a set and the matroid of linearly independent subsets of a vector space, respectively. Another natural matroid…
In this paper we introduce a new type of code, called projective nested cartesian code. It is obtained by the evaluation of homogeneous polynomials of a fixed degree on a certain subset of $\mathbb{P}^n(\mathbb{F}_q)$, and they may be seen…
At present, practical application and theoretical discussion of rough sets are two hot problems in computer science. The core concepts of rough set theory are upper and lower approximation operators based on equivalence relations. Matroid,…
We define almost affine vector rank-metric codes as subsets $\mathcal{C}\subseteq \mathbb{F}_{q^m}^n$ whose canonical projections have cardinalities that are powers of $q^m$, and prove that they naturally induce $q$-matroids. We establish…
This paper investigates the question of uniqueness of the reduced oriented matroid structure arising from root systems of a Coxeter group in real vector spaces. We settle the question for finite Coxeter groups, irreducible affine Weyl…