| Вестник ОмГУ | Выпуск | Тематика | Литература |
| Вестник Омского университета, 1997, Вып. 1. С. 12-13. © Омский государственный университет, 1997 |
УДК 519.4 |
A.V. Borovik
Department of Mathematics, UMIST,
PO Box 88, Manchester M60 1QD,
United Kingdom
Получена 3 августа 1996 г.
| Матроидные отображения матроидов Кокстера охарактеризованы в терминах слоений. |
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].
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
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 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 .
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
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 
| [1] | 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. |
| [2] | 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. |
| [3] | Gale D., Optimal assignments in an ordered set: an application of matroid theory // J. Combinatorial Theory. 1968. V.4. P.1073-1082. |
| [4] | Gelfand I.M., Serganova V.V. Combinatorial geometries and torus strata on homogeneous compact manifolds // Russian Math. Surveys. 1987. V.42. P.133-168. |
| [5] | Ronan M. Lectures on Buildings - Academic Press. Boston. 1989. |
| [6] | Tits J. A local approach to buildings, in The Geometric Vein (Coxeter Festschrift) Springer-Verlag, New York a.o., 1981. P.317-322. |