# intervalBruhat(WeylGroupElement,WeylGroupElement) -- elements between two given ones for the Bruhat order on a Weyl group

## Synopsis

• Function: intervalBruhat
• Usage:
intervalBruhat(u,v)
• Inputs:
• Outputs:
• an instance of the type HasseDiagram, of elements smaller than u and bigger than v together with the reflections to go from an element to the next one (on the right) attached to the corresponding edge.

## Description

 i1 : R=rootSystemA(3) o1 = RootSystem{...8...} o1 : RootSystem i2 : w1 = reduce(R,{2,1,2}) o2 = WeylGroupElement{RootSystem{...8...}, | -1 |} | -1 | | 3 | o2 : WeylGroupElement i3 : w2 = reduce(R,{1,2,1,3,2}) o3 = WeylGroupElement{RootSystem{...8...}, | -1 |} | -2 | | 1 | o3 : WeylGroupElement i4 : myInterval=intervalBruhat(w1,w2) o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {1, | 0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 0 |}}}, {WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {}}}} | -2 | | 2 | | -1 | | -1 | | -1 | | -3 | | 1 | | -1 | | 1 | | -1 | | 2 | | 2 | | 2 | | 1 | | 1 | | 3 | o4 : HasseDiagram

Each row of the Hasse diagram contains the elements of a certain length together with their links to the next row.

 i5 : myInterval#1 o5 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 0 |}}}, | -1 | | -1 | | 2 | | 2 | ------------------------------------------------------------------------ {WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | -1 |}}}} | -3 | | 1 | | 1 | | 1 | o5 : List