499 research outputs found
The Expected Codimension of a Matroid Variety
Matroid varieties are the closures in the Grassmannian of sets of points
defined by specifying which Pl\"ucker coordinates vanish and which don't. In
general these varieties are very ill-behaved, but in many cases one can
estimate their codimension by keeping careful track of the conditions imposed
by the vanishing of each Pl\"ucker coordinates on the columns of the matrix
representing a point of the Grassmannian. This paper presents a way to make
this procedure precise, producing a number for each matroid variety called its
expected codimension that can be computed combinatorially solely from the list
of Pl\"ucker coordinates that are prescribed to vanish. We prove that for a
special, well-studied class of matroid varieties called positroid varieties,
the expected codimension coincides with the actual codimension.Comment: 17 pages, 2 figure
Infinite Matroids and Determinacy of Games
Solving a problem of Diestel and Pott, we construct a large class of infinite
matroids. These can be used to provide counterexamples against the natural
extension of the Well-quasi-ordering-Conjecture to infinite matroids and to
show that the class of planar infinite matroids does not have a universal
matroid.
The existence of these matroids has a connection to Set Theory in that it
corresponds to the Determinacy of certain games. To show that our construction
gives matroids, we introduce a new very simple axiomatization of the class of
countable tame matroids
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
Infinite graphic matroids Part I
An infinite matroid is graphic if all of its finite minors are graphic and
the intersection of any circuit with any cocircuit is finite. We show that a
matroid is graphic if and only if it can be represented by a graph-like
topological space: that is, a graph-like space in the sense of Thomassen and
Vella. This extends Tutte's characterization of finite graphic matroids.
The representation we construct has many pleasant topological properties.
Working in the representing space, we prove that any circuit in a 3-connected
graphic matroid is countable
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