# Bruhat Order

The Bruhat order is defined on a Coxeter group $G$ in the following manner for $x,y\in G$:

1. $x\stackrel{t}{\to}y\Leftrightarrow x^{-1}y=t$ with $t$ a reflection (conjugate of a generator) and $\ell(x)<\ell(y)$,
2. $x\leq y\Leftrightarrow x\stackrel{t_0}{\to}x_1\stackrel{t_1}{\to}\cdots\stackrel{t_{n-1}}{\to}x_n\stackrel{t_n}{\to}y.$

Reflexivity is clear since $x\stackrel{1}{\to}x$ (assuming $1$ is considered a generator).  Transitivity is trivial.  And if we have $x\leq y$ and $y\leq x,$ then in the simple case where $x\stackrel{t}{\to}y$ and $y\stackrel{t'}{\to}x,$ we have

$x^{-1}y=t$

and

$y^{-1}x=t'$

which imply that $t'=t^{-1}$ and hence that $t=t'.$  So $x^{-1}y=y^{-1}x.$  Since we must have $\ell(x)=\ell(y),$ it follows that $x=y.$  The general cases are done by induction.  So we have antisymmetry.

So it is a partial order on the Coxeter group.  The Bruhat digraph is constructed with elements as vertices and a directed edge from $x$ to $y$ iff $x\leq y.$  Note how this ordering differs from the simplicial ordering we previously mentioned on the induced Coxeter complex.

Recall the notion of standard subgroups (also known as parabolic subgroups) of a Coxeter group where $J$ triply represents a subset of indices of a generating set, the corresponding generators indexed by those indices, and the subgroup generated by such generators.  Also recall the definition of descents of elements.  In the following definition, $I,J$ are used in the second sense (as subsets of generators), and we will use $G_J$ to represent the subgroup generated by $J.$

Definition 1.  Let $I\subseteq J\subseteq S$ and define

$\begin{array}{l}D_I^J=\{g\in G:I\subseteq D_R(g)\subseteq J\}\\D_I=D_I^I\\D^I=D_\varnothing^{S-I}.\end{array}$

The sets $D_I^J$ are called the right descent classes (or the analogous definition for left descent classes).

Proposition 2.  Let $J\subseteq S.$  Then every $g\in G$ has a unique factorization of the form $g=g^Jg_J$ where $g^J\in D^J$ and $g_J\in G_J.$  Moreover we have $\ell(g)=\ell(g^J)+\ell(g_J).$

We can define the map $P^J:G\to G^J$ where $P^J(g)=g^J.$  This map is clearly idempotent.  Moreover we have the following.

Proposition 3.  The map $P^J$ preserves Bruhat order $(g_1\leq g_2\Rightarrow g_1^J\leq g_2^J).$

[1]  Björner, Anders and Francesco Brenti.  Combinatorics of Coxeter Groups.  Vol. 231.  Graduate Texts in Mathematics.  Springer.  2005.