A.V. Borovik, Department of Mathematics, UMIST
1. Notation
This paper continues the works [1,2] and uses, with some modification, their terminology and notation. Throughout the paper W is a Coxeter group (possibly infinite) and P a finite standard parabolic subgroup of W. We identify the Coxeter group W with its Coxeter complex and refer to elements of W as chambers, to cosets with respect to a parabolic subgroup as residues, etc. We shall use the calligraphic letter  as a notation for the Coxeter complex of W and the symbol  for the set of left cosets of the parabolic subgroup P. We shall use the Bruhat ordering on  in its geometric interpretation, as defined in [2, Theorem 5.7]. The w-Bruhat ordering on is denoted by the same symbol  as the w-Bruhat ordering on . Notation  , <w, >w has obvious meaning.
We refer to Tits [6] or Ronan [5] for definitions of chamber systems, galleries, geodesic galleries, residues, panels, walls, half-complexes. A short review of these concepts can be also found in [1,2].
2. Coxeter matroids
If W is a finite Coxeter group, a subset  is called a Coxeter matroid (for W and P) if it satisfies the maximality property: for every  the set  contains a unique w-maximal element A; this means that  for all  . If  is a Coxeter matroid we shall refer to its elements as bases. Ordinary matroids constitute a special case of Coxeter matroids, for W=Symn and P the stabiliser in W of the set  [4]. The maximality property in this case is nothing else but the well-known optimal property of matroids first discovered by Gale [3].
In the case of infinite groups W we need to slightly modify the definition. In this situation the primary notion is that of a matroid map
i.e. a map satisfying the matroid inequality
The image  of  obviously satisfies the maximality property. Notice that, given a set with the maximality property, we can introduce the map by setting  be equal to the w-maximal element of . Obviously, is a matroid map. In infinite Coxeter groups the image  of the matroid map associated with a set satisfying the maximality property may happen to be a proper subset of (the set of all `extreme' or `corner' chambers of ; for example, take for a large rectangular block of chambers in the affine Coxeter group  ). This never happens, however, in finite Coxeter groups, where  .
So we shall call a subset  a matroid if satisfies the maximality property and every element of is w-maximal in with respect to some . After that we have a natural one-to-one correspondence between matroid maps and matroid sets.
We can assign to every Coxeter matroid for W and P the Coxeter matroid for W and 1 (or W-matroid).
Теорема 1. [2, Lemma 5.15] A map
is a matroid map if and only if the map
defined by  is also a matroid map.
Recall that  denotes the w-maximal element in the residue . Its existence, under the assumption that the parabolic subgroup P is finite, is shown in [2, Lemma 5.14].
In is a matroid map, the map  is called the underlying flag matroid map for and its image  the underlying flag matroid for the Coxeter matroid  . If the group W is finite then every chamber x of every residue  is w-maximal in for w the opposite to x chamber of and  , as a subset of the group W, is simply the union of left cosets of P belonging to .
3. Characterisation of matroid maps
Two subsets A and B in are called adjacent if there are two adjacent chambers  and  , the common panel of a and b being called a common panel of A and B.
Лемма 1. If A and B are two adjacent convex subsets of then all their common panels belong to the same wall  .
We say in this situation that is the common wall of A and B.
For further development of our theory we need some structural results on Coxeter matroids.
Теорема 2. A map  is a matroid map if and only if the following two conditions are satisfied.
(1) All the fibres  ,  , are convex subsets of .
(2) If two fibres and  of are adjacent then their images A and B are symmetric with respect to the wall containing the common panels of and  , and the residues A and B lie on the opposite sides of the wall from the sets , , correspondingly.
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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