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

Synopsis

• Function: underBruhat
• Usage:
underBruhat(B)
• Inputs:
• B, , of WeylGroupElement
• Outputs:
• a list, of elements smaller than the ones in B and of length one less 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=underBruhat(longestWeylGroupElement(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 : underBruhat(L1) o4 = {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, | 0 |}, {2, | -1 | 2 | | -1 | | 1 | -3 | | 2 | | 1 ------------------------------------------------------------------------ |}}}, {WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 0 |}, {2, | | -3 | | -1 | | | 1 | | 2 | ------------------------------------------------------------------------ | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | 2 |}, {{1, | -1 |}, | -1 | | -1 | | 1 | | 0 | | -2 | | 1 | ------------------------------------------------------------------------ {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, | 1 | 2 | | 2 | | 1 | -1 | | -1 | | -1 ------------------------------------------------------------------------ |}, {1, | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | | -1 | | -1 | | | 0 | | 2 | ------------------------------------------------------------------------ | -1 |}, {1, | 1 |}}}} | 2 | | 1 | | -1 | | -1 | o4 : List