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Matroid maps

Матроидные отображения матроидов Кокстера охарактеризованы в терминах слоений.

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.

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