# aboveBruhat(BasicList) -- The Weyl group elements just under the ones in the list for the Bruhat order

## Synopsis

• Usage:
aboveBruhat(B)
• Function: aboveBruhat
• Inputs:
• B, , of WeylGroupElement
• Outputs:
• a list, of elements greater than the ones in B and of length one more together with the root of the reflection composed with (on the right) to obtain the element

## Description

 ```i1 : R=rootSystemA(3) o1 = RootSystem{...8...} o1 : RootSystem``` ```i2 : L=aboveBruhat(neutralWeylGroupElement(R)) o2 = {{WeylGroupElement{RootSystem{...8...}, | -1 |}, | 2 |}, | 2 | | -1 | | 1 | | 0 | ------------------------------------------------------------------------ {WeylGroupElement{RootSystem{...8...}, | 2 |}, | -1 |}, | -1 | | 2 | | 2 | | -1 | ------------------------------------------------------------------------ {WeylGroupElement{RootSystem{...8...}, | 1 |}, | 0 |}} | 2 | | -1 | | -1 | | 2 | o2 : List``` ```i3 : L1=apply(L,x->x#0) o3 = {WeylGroupElement{RootSystem{...8...}, | -1 |}, | 2 | | 1 | ------------------------------------------------------------------------ WeylGroupElement{RootSystem{...8...}, | 2 |}, | -1 | | 2 | ------------------------------------------------------------------------ WeylGroupElement{RootSystem{...8...}, | 1 |}} | 2 | | -1 | o3 : List``` ```i4 : aboveBruhat(L1) o4 = {{WeylGroupElement{RootSystem{...8...}, | 2 |}, {{1, | -1 |}, {2, | -1 | 1 | | 1 | | 2 | -2 | | 1 | | -1 ------------------------------------------------------------------------ |}}}, {WeylGroupElement{RootSystem{...8...}, | 3 |}, {{1, | 0 |}, {2, | | -2 | | -1 | | | 1 | | 2 | ------------------------------------------------------------------------ | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, | 1 | | 1 | | 2 | | 1 | | 2 | | -1 | ------------------------------------------------------------------------ {1, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 1 | 1 | | -2 | | 1 | -1 | | 3 | | -1 ------------------------------------------------------------------------ |}, {1, | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | | -1 | | 3 | | | 0 | | -1 | ------------------------------------------------------------------------ | 0 |}, {2, | 2 |}}}} | -1 | | -1 | | 2 | | 0 | o4 : List```