1,854 research outputs found
Lagrangian Pairs and Lagrangian Orthogonal Matroids
Represented Coxeter matroids of types and , that is, symplectic
and orthogonal matroids arising from totally isotropic subspaces of symplectic
or (even-dimensional) orthogonal spaces, may also be represented in buildings
of type and , respectively. Indeed, the particular buildings
involved are those arising from the flags or oriflammes, respectively, of
totally isotropic subspaces. There are also buildings of type arising
from flags of totally isotropic subspaces in odd-dimensional orthogonal space.
Coxeter matroids of type are the same as those of type (since they
depend only upon the reflection group, not the root system). However, buildings
of type are distinct from those of the other types. The matroids
representable in odd dimensional orthogonal space (and therefore in the
building of type ) turn out to be a special case of symplectic (flag)
matroids, those whose top component, or Lagrangian matroid, is a union of two
Lagrangian orthogonal matroids. These two matroids are called a Lagrangian
pair, and they are the combinatorial manifestation of the ``fork'' at the top
of an oriflamme (or of the fork at the end of the Coxeter diagram of ).
Here we give a number of equivalent characterizations of Lagrangian pairs,
and prove some rather strong properties of them.Comment: Requires amssymb.sty; 12 pages, 2 LaTeX figure
Matroid toric ideals: complete intersection, minors and minimal systems of generators
In this paper, we investigate three problems concerning the toric ideal
associated to a matroid. Firstly, we list all matroids such that
its corresponding toric ideal is a complete intersection.
Secondly, we handle with the problem of detecting minors of a matroid from a minimal set of binomial generators of . In
particular, given a minimal set of binomial generators of we
provide a necessary condition for to have a minor isomorphic to
for . This condition is proved to be sufficient
for (leading to a criterion for determining whether is
binary) and for . Finally, we characterize all matroids
such that has a unique minimal set of binomial generators.Comment: 9 page
- …