14 research outputs found
Isotropical Linear Spaces and Valuated Delta-Matroids
The spinor variety is cut out by the quadratic Wick relations among the
principal Pfaffians of an n x n skew-symmetric matrix. Its points correspond to
n-dimensional isotropic subspaces of a 2n-dimensional vector space. In this
paper we tropicalize this picture, and we develop a combinatorial theory of
tropical Wick vectors and tropical linear spaces that are tropically isotropic.
We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid
polytopes, and we examine to what extent the Wick relations form a tropical
basis. Our theory generalizes several results for tropical linear spaces and
valuated matroids to the class of Coxeter matroids of type D
Polytopal and structural aspects of matroids and related objects
PhDThis thesis consists of three self-contained but related parts. The rst is focussed on
polymatroids, these being a natural generalisation of matroids. The Tutte polynomial is
one of the most important and well-known graph polynomials, and also features prominently
in matroid theory. It is however not directly applicable to polymatroids. For
instance, deletion-contraction properties do not hold. We construct a polynomial for
polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact
contains the same information as the Tutte polynomial when we restrict to matroids.
The second section is concerned with split matroids, a class of matroids which arises by
putting conditions on the system of split hyperplanes of the matroid base polytope. We
describe these conditions in terms of structural properties of the matroid, and use this
to give an excluded minor characterisation of the class.
In the nal section, we investigate the structure of clutters. A clutter consists of a nite
set and a collection of pairwise incomparable subsets. Clutters are natural generalisations
of matroids, and they have similar operations of deletion and contraction. We introduce
a notion of connectivity for clutters that generalises that of connectivity for matroids.
We prove a splitter theorem for connected clutters that has the splitter theorem for
connected matroids as a special case: if M and N are connected clutters, and N is a
proper minor of M, then there is an element in E(M) that can be deleted or contracted
to produce a connected clutter with N as a minor