Proceedings of Symposia in Pure Mathematics
Volume 56 (1994), Part 1
S u r l e s D e c o m p o s i t i o n s C e l l u l a i r e s d e s E s p a c e s G/B
This paper by C. Chevalley is a famous unpublished manuscript, not dated
but most likely from 1958, contemporary with the second volume of Cheval-
ley's seminar [CI] and [C2]. One of the results (see II below) is, in fact,
announced at the end of Chevalley's address at the 1958 ICM in Edinburgh
[C3]. Although the original manuscript was obviously not meant to be a final
version, Chevalley lent it to someone and this was the origin of a limited un-
derground distribution, so that early on quite a number of people had seen
a copy or were aware of its contents. The results became part of the official
literature in 1974, after the publication of [D], where M. Demazure provides
the statements and full proofs of all the results in Chevalley's paper.
In the title, G is a connected semi-simple algebraic group defined over
an algebraically closed groundfield k and B a Borel subgroup. Let T be a
maximal torus in B, W = N(T)/T the Weyl group of G with respect to
T. This translates e(w) (w e W) of the origin in G/B by (representatives
of) elements of W are the fixed points of T and their orbits Be(w) under
B are the Bruhat cells, isomorphic to affine spaces, and define a cellular
decomposition of G/B, the "Bruhat decomposition" of G/B. The paper is
mainly devoted to two problems pertaining to the Zariski closures X{w) =
B.e(w) of the B.e(w), which are now usually called the Schubert varieties.
(I) Chevalley first gives the now familiar combinatorial criterion for inclu-
sion of Schubert varieties:
c X(w) if and only if a reduced expression
for w' can be obtained from one for w by erasing some elements in it. This
is usually expressed by saying that w' ^ w in the "Bruhat ordering", a to-
tally unjustified terminology, unfortunately. Bruhat stated that publicly at
the Symposium in honor of his and Tits' 60th years. If a proper name has to
be attached to this partial ordering, it should be that of Chevalley. If, in view
of the abundant use in the literature of "Bruhat ordering", it is felt it would
1991 Mathematics Subject Classification. Primary 14M15; Secondary 14C15, 20G15.
©1994 American Mathematical Society
0082-0717/94 $1.00+ $.25 per page
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