1,111 research outputs found
Lagrangian Matroids: Representations of Type
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.Comment: Requires amssymb.sty; 17 page
Positively oriented matroids are realizable
We prove da Silva's 1987 conjecture that any positively oriented matroid is a
positroid; that is, it can be realized by a set of vectors in a real vector
space. It follows from this result and a result of the third author that the
positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a
closed ball.Comment: 20 pages, 3 figures, references adde
Oriented Matroids -- Combinatorial Structures Underlying Loop Quantum Gravity
We analyze combinatorial structures which play a central role in determining
spectral properties of the volume operator in loop quantum gravity (LQG). These
structures encode geometrical information of the embedding of arbitrary valence
vertices of a graph in 3-dimensional Riemannian space, and can be represented
by sign strings containing relative orientations of embedded edges. We
demonstrate that these signature factors are a special representation of the
general mathematical concept of an oriented matroid. Moreover, we show that
oriented matroids can also be used to describe the topology (connectedness) of
directed graphs. Hence the mathematical methods developed for oriented matroids
can be applied to the difficult combinatorics of embedded graphs underlying the
construction of LQG. As a first application we revisit the analysis of [4-5],
and find that enumeration of all possible sign configurations used there is
equivalent to enumerating all realizable oriented matroids of rank 3, and thus
can be greatly simplified. We find that for 7-valent vertices having no
coplanar triples of edge tangents, the smallest non-zero eigenvalue of the
volume spectrum does not grow as one increases the maximum spin \jmax at the
vertex, for any orientation of the edge tangents. This indicates that, in
contrast to the area operator, considering large \jmax does not necessarily
imply large volume eigenvalues. In addition we give an outlook to possible
starting points for rewriting the combinatorics of LQG in terms of oriented
matroids.Comment: 43 pages, 26 figures, LaTeX. Version published in CQG. Typos
corrected, presentation slightly extende
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{\"o}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.Comment: 21 pages, 4 figure
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