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Matroid maps

Matroid maps

Matroid maps

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  Matroid mapsas a notation for the Coxeter complex of W and the symbol  Matroid mapsfor the set of left cosets of the parabolic subgroup P. We shall use the Bruhat ordering on  Matroid mapsin its geometric interpretation, as defined in [2, Theorem 5.7]. The w-Bruhat ordering on  Matroid mapsis denoted by the same symbol  Matroid mapsas the w-Bruhat ordering on  Matroid maps. Notation  Matroid maps, <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  Matroid mapsis called a Coxeter matroid (for W and P) if it satisfies the maximality property: for every  Matroid mapsthe set  Matroid mapscontains a unique w-maximal element A; this means that  Matroid mapsfor all  Matroid maps. If  Matroid mapsis 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  Matroid maps[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

 Matroid maps

i.e. a map satisfying the matroid inequality

 Matroid maps

The image  Matroid mapsof  Matroid mapsobviously satisfies the maximality property. Notice that, given a set  Matroid mapswith the maximality property, we can introduce the map  Matroid mapsby setting  Matroid mapsbe equal to the w-maximal element of  Matroid maps. Obviously,  Matroid mapsis a matroid map. In infinite Coxeter groups the image  Matroid mapsof the matroid map associated with a set  Matroid mapssatisfying the maximality property may happen to be a proper subset of  Matroid maps(the set of all `extreme' or `corner' chambers of  Matroid maps; for example, take for  Matroid mapsa large rectangular block of chambers in the affine Coxeter group  Matroid maps). This never happens, however, in finite Coxeter groups, where  Matroid maps.

So we shall call a subset  Matroid mapsa matroid if  Matroid mapssatisfies the maximality property and every element of  Matroid mapsis w-maximal in  Matroid mapswith respect to some  Matroid maps. After that we have a natural one-to-one correspondence between matroid maps and matroid sets.

We can assign to every Coxeter matroid  Matroid mapsfor W and P the Coxeter matroid for W and 1 (or W-matroid).

Теорема 1. [2, Lemma 5.15] A map

 Matroid maps

is a matroid map if and only if the map

 Matroid maps

defined by  Matroid mapsis also a matroid map.

Recall that  Matroid mapsdenotes the w-maximal element in the residue  Matroid maps. Its existence, under the assumption that the parabolic subgroup P is finite, is shown in [2, Lemma 5.14].

In  Matroid mapsis a matroid map, the map  Matroid mapsis called the underlying flag matroid map for  Matroid mapsand its image  Matroid mapsthe underlying flag matroid for the Coxeter matroid  Matroid maps. If the group W is finite then every chamber x of every residue  Matroid mapsis w-maximal in  Matroid mapsfor w the opposite to x chamber of  Matroid mapsand  Matroid maps, as a subset of the group W, is simply the union of left cosets of P belonging to  Matroid maps.

3. Characterisation of matroid maps

Two subsets A and B in  Matroid mapsare called adjacent if there are two adjacent chambers  Matroid mapsand  Matroid maps, 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  Matroid mapsthen all their common panels belong to the same wall  Matroid maps.

We say in this situation that  Matroid mapsis 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  Matroid mapsis a matroid map if and only if the following two conditions are satisfied.

(1) All the fibres  Matroid maps,  Matroid maps, are convex subsets of  Matroid maps.

(2) If two fibres  Matroid mapsand  Matroid mapsof  Matroid mapsare adjacent then their images A and B are symmetric with respect to the wall  Matroid mapscontaining the common panels of  Matroid mapsand  Matroid maps, and the residues A and B lie on the opposite sides of the wall  Matroid mapsfrom the sets  Matroid maps,  Matroid maps, correspondingly.

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

Assume now that  Matroid mapssatisfies the conditions (1) and (2).

First we introduce, for any two adjacent fibbers  Matroid mapsand  Matroid mapsof the map  Matroid maps, the wall  Matroid mapsseparating them. Let  Matroid mapsbe the set of all walls  Matroid maps.

Now take two arbitrary residues  Matroid mapsand chambers  Matroid mapsand  Matroid maps. We wish to prove  Matroid maps.

Consider a geodesic gallery

 Matroid maps

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

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.

Для подготовки данной работы были использованы материалы с сайта http://www.omsu.omskreg.ru/




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