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

## Synopsis

• Function: intervalBruhat
• Usage:
intervalBruhat(u,v)
• Inputs:
• Outputs:
• an instance of the type HasseDiagram, of elements of minimal length in their coset, bigger than u and smaller 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 : P=parabolic(R,set {3}) o2 = set {3} o2 : Parabolic i3 : w1 = reduce(R,{2}) o3 = WeylGroupElement{RootSystem{...8...}, | 2 |} | -1 | | 2 | o3 : WeylGroupElement i4 : w2 = reduce(R,{1,2,1,3,2}) o4 = WeylGroupElement{RootSystem{...8...}, | -1 |} | -2 | | 1 | o4 : WeylGroupElement i5 : myInterval=intervalBruhat(P % w1,P % w2) o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {1, | 0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 0 |}, {1, | -1 |}, {2, | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | -1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | 2 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | 2 |}, {{1, | 0 |}, {2, | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, | 0 |}, {2, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | 3 |}, {{0, | 0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 2 |}, {}}}} | -2 | | 2 | | -1 | | -1 | | -1 | | 1 | | -1 | | -3 | | 1 | | 2 | | -1 | | -1 | | 2 | | -3 | | -1 | | -1 | | 1 | | -1 | | 0 | | 1 | | 1 | | -2 | | -1 | | -2 | | -1 | | -1 | | 1 | | -1 | | 2 | | 2 | | 2 | | 1 | | 0 | | 1 | | 1 | | -1 | | 3 | | 0 | | -1 | | 2 | | 2 | | 0 | | 1 | | 2 | | 1 | | 2 | | -1 | | 3 | | 0 | | 1 | | 2 | | 2 | o5 : HasseDiagram

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

 i6 : myInterval#1 o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 0 |}, {1, | -1 | -1 | | -1 | | 1 | 2 | | 2 | | 1 ------------------------------------------------------------------------ |}, {2, | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | | -1 | | -3 | | | 0 | | 1 | ------------------------------------------------------------------------ | -1 |}, {1, | -1 |}}}} | 1 | | 2 | | 1 | | -1 | o6 : List