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symmetricGroupPoset -- constructs the poset of the symmetric group under a partial order

Description

Using the Posets package, we can construct the poset on the symmetric group $\mathfrak{S}_n$ under a partial order. See [BB05] for more details on Bruhat order.

i1 : symmetricGroupPoset(4, strongBruhatOrder)

o1 = Relation Matrix: | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 |
                      | 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1 1 1 0 1 0 1 1 1 |
                      | 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 |
                      | 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 |
                      | 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

o1 : Poset

With a little bit of finagling, we can also construct the poset using the non-default left weak Bruhat order.

i2 : partialOrder = (w,v) -> weakBruhatOrder(w,v, Side=>"left")

o2 = partialOrder

o2 : FunctionClosure
i3 : symmetricGroupPoset(4, partialOrder)

o3 = Relation Matrix: | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
                      | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 |
                      | 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 0 1 1 |
                      | 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 |
                      | 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 |
                      | 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 |
                      | 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 1 |
                      | 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 |
                      | 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 |
                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

o3 : Poset

See also

Ways to use symmetricGroupPoset:

  • symmetricGroupPoset(ZZ,Function)

For the programmer

The object symmetricGroupPoset is a method function.


The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/Permutations/Documentation/mainDocs.m2:746:0.