Bruhat order
Template:Short description 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 Template:Harvtxt, and the analogue for more general semisimple algebraic groups was studied by Template:Harvtxt. Template:Harvtxt 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 Template:Harvs.
Definition
If Template:Math is a Coxeter system with generators Template:Mvar, then the Bruhat order is a partial order on the group Template:Mvar. The definition of Bruhat order relies on several other definitions: first, reduced word for an element Template:Mvar of Template:Mvar is a minimum-length expression of Template:Mvar as a product of elements of Template:Mvar, and the length Template:Math of Template:Mvar is the length of its reduced words. Then the (strong) Bruhat order is defined by Template:Math if some substring of some (or every) reduced word for Template:Mvar is a reduced word for Template:Mvar. (Here a substring is not necessarily a consecutive substring.)
There are two other related partial orders:
- the weak left (Bruhat) order is defined by Template:Math if some final substring of some reduced word for Template:Mvar is a reduced word for Template:Mvar, and
- the weak right (Bruhat) order is defined by Template:Math if some initial substring of some reduced word for Template:Mvar is a reduced word for Template:Mvar.
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 Template:Math whenever Template:Math for some reflection Template:Mvar and Template:Math. 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 , and thus this poset is Eulerian, meaning its Möbius function is produced by the rank function on the poset.