In mathematics, the Bruhat order (also called strong order or strong Bruhat order or Chevalley order or Bruhat–Chevalley order or Chevalley–Bruhat order) is a partial order on the elements of a Coxeter group

Coxeter group

In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

, that corresponds to the inclusion order on Schubert varieties.

History

The Bruhat order on the Schubert varieties of a flag manifold or 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

Bruhat decomposition

In mathematics, the Bruhat decomposition G = BWB into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases...

 introduced by François Bruhat

François Bruhat

François Georges René Bruhat was a French mathematician who worked on algebraic groups. The Bruhat order of a Weyl group, the Bruhat decomposition, and the Schwartz–Bruhat functions are named after him....

.

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.

• The weak left (Bruhat) order is defined by u≤_Lv 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≤_Rv if some initial substring of some reduced word for v is a reduced word for u.

Bruhat graph

The Bruhat graph is a directed graph that is distinctly 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 and l(u)≤l(v).