Matroid maps

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Доказательство. If is a matroid map then the satisfaction of conditions (1) and (2) is the main result of [2].

Assume now that satisfies the conditions (1) and (2).

First we introduce, for any two adjacent fibbers and of the map , the wall separating them. Let be the set of all walls .

Now take two arbitrary residues and chambers and . We wish to prove .

Consider a geodesic gallery

connecting the chambers u and v. Let now the chamber x moves along from u to v, then the corresponding residue moves from to . Since the geodesic gallery intersects every wall no more than once [5, Lemma 2.5], the chamber x crosses each wall in no more than once and, if it crosses , it moves from the same side of as u to the opposite side. But, by the assumptions of the theorem, this means that the residue crosses each wall no more than once and moves from the side of opposite u to the side containing u. But, by the geometric interpretation of the Bruhat order, this means [2, Theorem 5.7] that decreases, with respect to the u-Bruhat order, at every such step, and we ultimately obtain

Список литературы

Borovik A.V., Gelfand I.M. WP-matroids and thin Schubert cells on Tits systems // Advances Math. 1994. V.103. N.1. P.162-179.

Borovik A.V., Roberts K.S. Coxeter groups and matroids, in Groups of Lie Type and Geometries, W. M. Kantor and L. Di Martino, eds. Cambridge University Press. Cambridge, 1995 (London Math. Soc. Lect. Notes Ser. V.207) P.13-34.

Gale D., Optimal assignments in an ordered set: an application of matroid theory // J. Combinatorial Theory. 1968. V.4. P.1073-1082.

Gelfand I.M., Serganova V.V. Combinatorial geometries and torus strata on homogeneous compact manifolds // Russian Math. Surveys. 1987. V.42. P.133-168.

Ronan M. Lectures on Buildings - Academic Press. Boston. 1989.

Tits J. A local approach to buildings, in The Geometric Vein (Coxeter Festschrift) Springer-Verlag, New York a.o., 1981. P.317-322.

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