Representations of Matroids in Semimodular Lattices

  title={Representations of Matroids in Semimodular Lattices},
  author={Alexandre V. Borovik and Israel M. Gelfand and Neil White},
  journal={Eur. J. Comb.},
Representations of matroids in semimodular lattices and Coxeter matroids in chamber systems are considered in this paper. 
The Topological Representation of Oriented Matroids
A new proof of the Topological Representation Theorem for oriented matroids in the general rank case uses hyperline sequences and the generalized Schonflies theorem and shows that one can read off orientedMatroids from arrangements of embedded spheres of codimension one, even if wild spheres are involved.
Matroids, hereditary collections and simplicial complexes having boolean representations
Inspired by the work of Izakhian and Rhodes, a theory of representation of hereditary collections by boolean matrices is developed. This corresponds to representation by finite $\vee$-generated
On the Topology of the Combinatorial Flag Varieties
We prove that the simplicial complex Ωn of chains of matroids (with respect to the ordering by the quotient relation) on n elements is shellable. This follows from a more general result on
Matroids and Coxeter groups
The paper describes a few ways in which the concept of a Coxeter group (in its most ubiquitous manifestation, the symmetric group) emerges in the theory of ordinary matroids: • Gale’s maximality
Flag matroids with coefficients
. This paper is a direct generalization of Baker-Bowler theory to flag matroids, including its moduli interpretation as developed by Baker and the second author for matroids. More explicitly, we
A Dual Fano, and Dual Non-Fano Matroidal Network
This paper examines the basic information on network coding and matroid theory, then goes over the method of creating matroidal networks, andconstructs matroidAL networks from the dual of the fano matroid and theDual of the non-fano matroid.
Combinatorial Flag Varieties


The geometry of the chamber system of a semimodular lattice
In this paper geometric properties of the following metric space C are studied. Its elements are called chambers and are the maximal chains of a semimodular lattice X of finite height and its metric
Combinatorial geometries and torus strata on homogeneous compact manifolds
CONTENTSIntroduction § 1. Torus orbits and strata on the Grassmannian § 2. Matroids and strata on the Grassmannian § 3. The moment mapping and toric varieties § 4. The geometry of compact homogeneous
Reflection groups and coxeter groups
Part I. Finite and Affine Reflection Groups: 1. Finite reflection groups 2. Classification of finite reflection groups 3. Polynomial invariants of finite reflection groups 4. Affine reflection groups
The gallery distance of flags
An explicit formula for the gallery distance of two maximal flags in a vector space is given. The main tool of the proof is the Jordan-Hölder permutation. The result and its proof hold more generally
Symplectic Matroids
A symplectic matroid is a collection B of k-element subsets of J = {1, 2, ..., n, 1*, 2*, ...; n*}, each of which contains not both of i and i* for every i ≤ n, and which has the additional property
Buildings of Spherical Type and Finite BN-Pairs
These notes are a slightly revised and extended version of mim- graphed notes written on the occasion of a seminar on buildings and BN-pairs held at Oberwolfach in April 1968. Their main purpose is
A Local Approach to Buildings
The object of this paper is the comparison of two notions of (combinatorial) buildings, that of [14] (or [1]), and an earlier version (cf. e.g. [10]), which has lately regained interest through the
Lectures on Buildings
The Geometric vein : the Coxeter Festschrift
H. S. M. Coxeter: Published Works.- I: Polytopes and Honeycombs.- Uniform Tilings with Hollow Tiles.- Spherical Tilings with Transitivity Properties.- Some Isonemal Fabrics on Polyhedral Surfaces.-