Annals of Combinatorics 1 (1997) 17-26

The Lattice of Flats and its Underlying Flag Matroid Polytope

Alexandre V. Borovik 1, Israel M. Gelfand2, Andrew Vince3, and Neil White3

1Department of Mathematics, UMIST, P.O. Box 88, Manchester M60 1QD, UK

2Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA

3Department of Mathematics, University of Florida, Gainesville, FL 32611, USA

Received October 3, 1996

AMS Subject Classification: 05B35, 05E15, 20F55, 52B40

Abstract. Let M be a matroid and F the collection of all linear orderings of bases of M, or flags of M. We define the flag matroid polytope Δ (F). We determine when two vertices of Δ (F) are adjacent, and provide a bijection between maximal chains in the lattice of flats of M and certain maximal faces of Δ (F).

Keywords: matroid, Coxeter matroid, flag matroid, flag matroid polytope


1.  R. Blind and P. Mani-Levitska, On puzzles and polytope isomorphisms, Aequationes Math. 34 (1987) 287–297.

2.  A. Borovik, M. Gelfand, and N. White, Exchange properties for Coxeter matroids and oriented matroids, submitted.

3.  A. Borovik, M. Gelfand, and N. White, Coxeter matroid polytopes, Ann. Combin., to appear.

4.  A. Borovik and K.S. Roberts, Coxeter groups and matroids, In: Groups of Lie Type and their Geometries, W. M. Kantor and L. Di Martino, Eds., Cambridge University Press, Cambridge, 1995, pp. 13–34.

5.  I.M. Gelfand, M. Goresky, R.D. MacPherson, and V.V. Serganova, Combinatorial geometries, convex polyhedra, and Schubert cells, Adv. Math. 63 (1987) 301–316.

6.  I.M. Gelfand and V.V. Serganova, Combinatorial geometries and torus strata on homogeneous compact manifolds, Russian Math. Surveys 42 (1987) 133–168. See also I.M. Gelfand, Collected Papers, Vol. III, Springer-Verlag, New York, 1989, pp. 926–958.

7.  J.G. Oxley, Matroid Theory, Oxford University Press, 1992.

8.  V.V. Serganova, A. Vince, and A. Zelevinsky, A geometric characterization of Coxeter matroids, Ann. Combin., to appear.

9.  N. White, Ed., Theory of Matroids, Cambridge University Press, 1986

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