Representations of Matroids in Semimodular Lattices

@article{Borovik2001RepresentationsOM,
  title={Representations of Matroids in Semimodular Lattices},
  author={Alexandre V. Borovik and Israel M. Gelfand and Neil White},
  journal={Eur. J. Comb.},
  year={2001},
  volume={22},
  pages={789-799}
}
Representations of matroids in semimodular lattices and Coxeter matroids in chamber systems are considered in this paper. 
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References

SHOWING 1-10 OF 26 REFERENCES
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.-
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2
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