3,103 research outputs found
On a matroid identity
The purpose of this note is to prove an identity for generalized Tutte-Grothendieck invariants, at least two special cases of which have already proved to be of considerable use. In addition, one of these special cases is used to strengthen results of Lindström on the critical exponent of a representable matroid and the chromatic number of a regular matroid. © 1983
Recipe theorem for the Tutte polynomial for matroids, renormalization group-like approach
Using a quantum field theory renormalization group-like differential
equation, we give a new proof of the recipe theorem for the Tutte polynomial
for matroids. The solution of such an equation is in fact given by some
appropriate characters of the Hopf algebra of isomorphic classes of matroids,
characters which are then related to the Tutte polynomial for matroids. This
Hopf algebraic approach also allows to prove, in a new way, a matroid Tutte
polynomial convolution formula appearing in W. Kook {\it et. al., J. Comb.
Series} {\bf B 76} (1999).Comment: 14 pages, 3 figure
On perturbations of highly connected dyadic matroids
Geelen, Gerards, and Whittle [3] announced the following result: let be a prime power, and let be a proper minor-closed class of
-representable matroids, which does not contain
for sufficiently high . There exist integers
such that every vertically -connected matroid in is a
rank- perturbation of a frame matroid or the dual of a frame matroid
over . They further announced a characterization of the
perturbations through the introduction of subfield templates and frame
templates.
We show a family of dyadic matroids that form a counterexample to this
result. We offer several weaker conjectures to replace the ones in [3], discuss
consequences for some published papers, and discuss the impact of these new
conjectures on the structure of frame templates.Comment: Version 3 has a new title and a few other minor corrections; 38
pages, including a 6-page Jupyter notebook that contains SageMath code and
that is also available in the ancillary file
A class of matroids derived from saturated chain partitions of partially ordered sets
AbstractThe simultaneously k- and (k − 1)-saturated chain partitions of a finite partially ordered set P determine a matroid Gk(P). This matroid is a gammoid. The identity on P induces a strong map from Gk(P) to Gk + 1(P). This strong map has a linear representation
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
Topological representations of matroid maps
The Topological Representation Theorem for (oriented) matroids states that
every (oriented) matroid can be realized as the intersection lattice of an
arrangement of codimension one homotopy spheres on a homotopy sphere. In this
paper, we use a construction of Engstr\"om to show that structure-preserving
maps between matroids induce topological mappings between their
representations; a result previously known only in the oriented case.
Specifically, we show that weak maps induce continuous maps and that the
process is a functor from the category of matroids with weak maps to the
homotopy category of topological spaces. We also give a new and conceptual
proof of a result regarding the Whitney numbers of the first kind of a matroid.Comment: Final version, 21 pages, 8 figures; Journal of Algebraic
Combinatorics, 201
Tensor structure from scalar Feynman matroids
We show how to interpret the scalar Feynman integrals which appear when
reducing tensor integrals as scalar Feynman integrals coming from certain nice
matroids.Comment: 12 pages, corrections suggested by referee
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