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  or Ronan 
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 . The maximality property in this
case is nothing else but the well-known optimal property of matroids first
discovered by Gale .
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
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).
1. [2, Lemma 5.15] A map
is a matroid map if and only if the
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
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 .
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 .
Assume now that satisfies the conditions (1) and
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 .
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
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Gale D., Optimal assignments in an
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Gelfand I.M., Serganova V.V.
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