Bruhat order

In mathematics, the Bruhat order (also called the strong order, strong Bruhat order, Chevalley order, Bruhat–Chevalley order, or Chevalley–Bruhat order) is a partial order on the elements of a Coxeter group, that corresponds to the inclusion order on Schubert varieties.

History
The Bruhat order on the Schubert varieties of a flag manifold or a Grassmannian was first studied by, and the analogue for more general semisimple algebraic groups was studied by. started the combinatorial study of the Bruhat order on the Weyl group, and introduced the name "Bruhat order" because of the relation to the Bruhat decomposition introduced by François Bruhat.

The left and right weak Bruhat orderings were studied by.

Definition
If (W, S) is a Coxeter system with generators S, then the Bruhat order is a partial order on the group W. Recall that a reduced word for an element w of W is a minimal length expression of w as a product of elements of S, and the length ℓ(w) of w is the length of a reduced word.


 * The (strong) Bruhat order is defined by u ≤ v if some substring of some (or every) reduced word for v is a reduced word for u. (Note that here a substring is not necessarily a consecutive substring.)


 * The weak left (Bruhat) order is defined by u ≤L v  if some final substring of some  reduced word for v is a reduced word for u.
 * The weak right (Bruhat) order is defined by u ≤R v  if some initial substring of some  reduced word for v is a reduced word for u.

For more on the weak orders, see the article weak order of permutations.

Bruhat graph
The Bruhat graph is a directed graph related to the (strong) Bruhat order. The vertex set is the set of elements of the Coxeter group and the edge set consists of directed edges (u, v) whenever u = tv for some reflection t and ℓ(u) < ℓ(v). One may view the graph as an edge-labeled directed graph with edge labels coming from the set of reflections. (One could also define the Bruhat graph using multiplication on the right; as graphs, the resulting objects are isomorphic, but the edge labelings are different.)

The strong Bruhat order on the symmetric group (permutations) has Möbius function given by $$\mu(\pi,\sigma)=(-1)^{\ell(\sigma)-\ell(\pi)}$$, and thus this poset is Eulerian, meaning its Möbius function is produced by the rank function on the poset.