The **Bruhat order** is defined on a Coxeter group in the following manner for :

- with a reflection (conjugate of a generator) and ,

Reflexivity is clear since (assuming is considered a generator). Transitivity is trivial. And if we have and then in the simple case where and we have

and

which imply that and hence that So Since we must have it follows that 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 to iff 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 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, are used in the second sense (as subsets of generators), and we will use to represent the subgroup generated by

**Definition 1.** Let and define

The sets are called the **right descent classes** (or the analogous definition for left descent classes).

**Proposition 2.** Let Then every has a unique factorization of the form where and Moreover we have

We can define the map where This map is clearly idempotent. Moreover we have the following.

**Proposition 3.** The map preserves Bruhat order

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