# Questions tagged [oriented-matroids]

Hyperplane arrangements, discrete geometry, convex polytopes, and optimization. For more general questions concerning matroids, use the matroid-theory tag.

16
questions

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### Every triangulation of an oriented matroid is partitionable

In his unpublished article, Combinatorial properties of triangulations of oriented matroids, Julian Webster proves that every triangulation of an oriented matroid is partitionable.
Does this result ...

**9**

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**0**answers

161 views

### How many components are there in the space of “generic” planar N-gons? (Mnev's revenge)

Call an ordered $N$-tuple of points in the Euclidean plane ${\mathbb R} ^2$ "in general position" if no three points of the points in the set are collinear. As a function of $N$ how many components ...

**3**

votes

**1**answer

126 views

### Matroids of hypercubes

Let $M_k$ be the (oriented) matroid of the $2^k$ points $B_k = \{-1, 1\}^k$ in $\mathbb R^k$. In other words, the (oriented) circuits of $M_k$ are the minimal (signed) linear dependences among $B_k$.
...

**5**

votes

**1**answer

166 views

### Reconstructing an oriented matroid from its deletion and contraction

Suppose that $\mathcal{M}_1$ and $\mathcal M_2$ are two oriented matroids on the same ground set $E$. Under what conditions on $\mathcal{M}_1$ and $\mathcal{M}_2$ is there an oriented matroid $\...

**6**

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151 views

### Matroid Representation of the Antichains of a Poset

Introduction
I am studying a problem in which the antichains of a poset are of key importance. They are naturally geometrically embedded as vectors in the space $\mathbb{R}^P$, where $P$ is the poset,...

**2**

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111 views

### Will the following construction leads to an counterexample of strong mapping conjecture on realizable oriented matroids?

The strong map conjecture asserts that any strong map $\mathcal{M}_1\rightarrow\mathcal{M}_2$ admit a factorization into an extension followed by a contraction. For which the counterexample has been ...

**9**

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158 views

### Is every planar point set the projection of vertices of a neighborly 4-polytope?

More exactly, written in coordinates, I'm curious to know if for every point set $(x_i,y_i)$ there are $(x_i,y_i,z_i,w_i)$ that are vertices of a neighborly polytope.
This problem comes from a simple ...

**6**

votes

**1**answer

450 views

### The Universality Theorem by Mnev for uniform oriented matroids of rank 4 and higher

According to the Universality Theorem by Mnev (see below theorem 8.6.6 from [1]), for any open semialgebraic variety V there is a uniform oriented matroid of rank 3 whose realization space is stably ...

**1**

vote

**0**answers

68 views

### What are / could be the applications of Delaunay oriented matroids?

The Delaunay oriented matroid is studied in detail by F. Santos (actually this paper is more general).
For a set of points $S$ (in any dimension), let $C$ be a sphere, $C^+$ be its interior, $C^-$ be ...

**6**

votes

**1**answer

349 views

### Small remarkable matroids

Working on phased matroids (a generalization of oriented matroid to the complex case) I've found an interesting formula for computing the inner Tutte group (and, hence, all the Tutte groups) ...

**5**

votes

**2**answers

156 views

### The lattice of covectors of an oriented matroid

Let $M$ be an oriented matroid on the ground set $E$, and let $L(M)$ be its ranked poset of covectors. By definition, $L(M)$ is a sub-poset of the poset $\{0, \pm 1\}^E$, ordered by putting $0 < \...

**3**

votes

**1**answer

437 views

### Positroids and Totally Nonnegative Complex Grassmanian

Recently I begin working on matroids, in particular to a generalization of oriented matroids to the complex case.
I found on arxiv the following interesting articles:
1)Alexander Postnikov: Total ...

**4**

votes

**1**answer

172 views

### Circuits in a linear oriented matroid

Given $E$ a finite subset of a real vector space $V$, a circuit of the associated matroid is a minimal linearly dependent subset of $E$. For each circuit $\underline C$, a minimal linear dependence
$$\...

**6**

votes

**2**answers

255 views

### Functionals on oriented matroids

Oriented matroids are abstractions of hyperplane arrangements, or equivalently vector configurations. Let me recall the definition in terms of covectors.
Let $R=\lbrace 0,+,-\rbrace$ with the monoid ...

**2**

votes

**1**answer

182 views

### Realizability of extensions of a free oriented matroid by an independent set

Question:
I am searching for a non-realizable matroid with few dependencies relative to the number of points. Precisely, I would like to find a non-realizable (over $\mathbb{R}$) oriented matroid $M$ ...

**9**

votes

**1**answer

698 views

### Triangulations of polytopes and tilings of zonotopes

Consider a set $A = \{ a_1,a_2,\ldots, a_n \} $ of vectors in $\mathbb{R}^d$, which lie in a common affine hyperplane. Two convex polytopes may be obtained from $A$, namely the convex hull of the ...