\magnification=1200
\hsize=4in
\overfullrule=0pt
\input amssym
\nopagenumbers
\noindent
%
%
{\bf John R. Stembridge}
%
%
\medskip
\noindent
%
%
{\bf Tight Quotients and Double Quotients in the Bruhat Order}
%
%
\vskip 5mm
\noindent
%
%
%
%
It is a well-known theorem of Deodhar that the Bruhat ordering of a
Coxeter group is the conjunction of its projections onto quotients
by maximal parabolic subgroups. Similarly, the Bruhat order is also
the conjunction of a larger number of simpler quotients obtained by
projecting onto two-sided (i.e., ``double'') quotients by pairs of
maximal parabolic subgroups. Each one-sided quotient may be represented
as an orbit in the reflection representation, and each double quotient
corresponds to the portion of an orbit on the positive side of certain
hyperplanes. In some cases, these orbit representations are ``tight''
in the sense that the root system induces an ordering on the orbit that
yields effective coordinates for the Bruhat order, and hence also
provides upper bounds for the order dimension. In this paper, we
(1) provide a general characterization of tightness for one-sided quotients,
(2) classify all tight one-sided quotients of finite Coxeter groups,
and (3) classify all tight double quotients of affine Weyl groups.
\bye