Lagrangian Matroids: Representations of Type BnB_n

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

On the weak order of Coxeter groups

This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of W to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general).Comment: 37 pages, submitte

Dessins, their delta-matroids and partial duals

Given a map $\mathcal M$ on a connected and closed orientable surface, the delta-matroid of $\mathcal M$ is a combinatorial object associated to $\mathcal M$ which captures some topological information of the embedding. We explore how delta-matroids associated to dessins d'enfants behave under the action of the absolute Galois group. Twists of delta-matroids are considered as well; they correspond to the recently introduced operation of partial duality of maps. Furthermore, we prove that every map has a partial dual defined over its field of moduli. A relationship between dessins, partial duals and tropical curves arising from the cartography groups of dessins is observed as well.Comment: 34 pages, 20 figures. Accepted for publication in the SIGMAP14 Conference Proceeding

Valuations for matroid polytope subdivisions

We prove that the ranks of the subsets and the activities of the bases of a matroid define valuations for the subdivisions of a matroid polytope into smaller matroid polytopes.Comment: 19 pages. 2 figures; added section 6 + other correction

Positively oriented matroids are realizable

We prove da Silva's 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result and a result of the third author that the positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a closed ball.Comment: 20 pages, 3 figures, references adde