hasseDiagramToGraph(HasseDiagram) -- turning a hasse diagram into a graph (intended for graphic representation)

i1 : R=rootSystemA(3)

o1 = RootSystem{...8...}

o1 : RootSystem

i2 : w1 = reduce(R,{2})

o2 = WeylGroupElement{RootSystem{...8...}, |  2 |}
                                           | -1 |
                                           |  2 |

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 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | -1 |}, {2, |  1 |}, {4, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, | -1 |}, {4, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1, | -1 |}, {3, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {3, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
                                                          | -2 |        | -1 |       |  1 |       |  2 |                                              | -3 |        |  1 |       |  2 |       |  1 |       | 0 |                                            |  2 |        | 0 |       |  2 |       | -1 |                                            | -1 |        | -1 |       |  1 |       | -1 |                                              | -1 |        |  2 |       | -1 |                                            | -3 |        | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            |  1 |        | 0 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
                                                          |  1 |        |  2 |       | -1 |       | -1 |                                              |  1 |        |  1 |       | -1 |       | -1 |       | 1 |                                            | -1 |        | 1 |       | -1 |       |  2 |                                            |  2 |        |  2 |       |  1 |       |  0 |                                              |  3 |        | -1 |       |  0 |                                            |  2 |        |  2 |       |  0 |                                            | -1 |        |  2 |       | -1 |                                            |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       |  1 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |

o4 : HasseDiagram

i5 : hasseDiagramToGraph(myInterval)

o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 2}, {, 4}}}, {, {{, 2}, {, 3}, {, 4}}}, {, {{, 0}, {, 1}, {, 3}}}}, {{, {{, 0}, {, 3}}}, {, {{, 0}, {, 2}}}, {, {{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}}, {{, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}

o5 : HasseGraph

i6 : hasseDiagramToGraph(myInterval,"labels"=>"reduced decomposition")

o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{232, 0}, {2, 1}, {121, 2}, {12321, 4}}}, {1232, {{12321, 2}, {2, 3}, {3, 4}}}, {1213, {{3, 0}, {232, 1}, {1, 3}}}}, {{121, {{2, 0}, {1, 3}}}, {213, {{3, 0}, {1, 2}}}, {232, {{3, 1}, {2, 2}}}, {123, {{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232, 3}}}}, {{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, {{2, {}}}}

o6 : HasseGraph