15 research outputs found
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
Lattice path matroids and quotients
We characterize the quotients among lattice path matroids (LPMs) in terms of
their diagrams. This characterization allows us to show that ordering LPMs by
quotients yields a graded poset, whose rank polynomial has the Narayana numbers
as coefficients.
Furthermore, we study full lattice path flag matroids and show that --
contrary to arbitrary positroid flag matroids -- they correspond to points in
the nonnegative flag variety. At the basis of this result lies an
identification of certain intervals of the strong Bruhat order with lattice
path flag matroids.
A recent conjecture of Mcalmon, Oh, and Xiang states a characterization of
quotients of positroids. We use our results to prove this conjecture in the
case of LPMs.Comment: 29 pages, 12 figures, 2 table