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

Function: intervalBruhat
Usage:

intervalBruhat(u,v)
Inputs:
- u, an instance of the type WeylGroupElement, the lower bound of the interval
- v, an instance of the type WeylGroupElement, the upper bound of the interval
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, |  0 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {}}}}
                                                          | -2 |        | -1 |       |  2 |                                              | -3 |        |  1 |                                            | -1 |        | -1 |                                              | -1 |
                                                          |  1 |        |  2 |       | -1 |                                              |  1 |        |  1 |                                            |  2 |        |  2 |                                              |  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...}, |  1 |}, {{0, | -1 |}}},
                                             | -3 |        |  1 |    
                                             |  1 |        |  1 |    
     ------------------------------------------------------------------------
     {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}}}}
                                            | -1 |        | -1 |
                                            |  2 |        |  2 |

o5 : List

Ways to use this method:

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

The source of this document is in WeylGroups.m2:4429:0.