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




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.


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