# bruhatOrder -- computes the Bruhat order on a generalized flag variety

## Synopsis

• Usage:
P = bruhatOrder(X)
• Inputs:
• X, , a generalized flag variety or Schubert variety
• Outputs:
• P, an instance of the type Poset, the Bruhat order

## Description

Given a GKMVariety $X$ created by either generalizedFlagVariety or generalizedSchubertVariety, computes and returns the Bruhat order corresponding to the Bruhat decomposition. The resulting poset is cached in $X$, and can be accessed by either cellOrder or bruhatOrder.

 i1 : Fl3 = generalizedFlagVariety("A",2,{1,2}) o1 = a GKM variety with an action of a 3-dimensional torus o1 : GKMVariety i2 : cellOrder Fl3 stdio:2:1:(3): error: no cell order defined on this moment graph i2 : P = bruhatOrder Fl3 o2 = P o2 : Poset i3 : #(coveringRelations P) == 8 o3 = true i4 : cellOrder Fl3 o4 = P o4 : Poset