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How to find the longest word in a subgroup of the symmetric group using Sage?

asked 4 years ago

lijr07 gravatar image

Let Sn be the symmetric group over {1,2,,n}. Let J be a subset of {1,,n1} and let WJ be the subgroup of Sn generated by sj,jJ{1,,n1}, where sj's are simple reflections. How to find the longest word in WJ in Sage? The following is some codes.

W = SymmetricGroup(8)
[s1,s2,s3,s4,s5,s6,s7] = W.simple_reflections()

Thank you very much.

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Something like the following...?!

sage: S8 = SymmetricGroup(8)                                                                                                                                                                          
sage: S7 = SymmetricGroup(7)                                                                                                                                                                          
sage: S8( S7.long_element() )                                                                                                                                                                         
(1,7)(2,6)(3,5)
dan_fulea gravatar imagedan_fulea ( 4 years ago )

@lijr07 could you please provide a definition of longest word ?

tmonteil gravatar imagetmonteil ( 4 years ago )

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answered 4 years ago

FrédéricC gravatar image

Here is a starting point. The rest is standard Coxeter theory.

sage: W = CoxeterGroup(['A',4])
sage: w = W.long_element()
sage: t = w.coset_representative([1,2])
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@FrédéricC, thank you very much. I tried to use the following commands to find the longest word in WJ, J={1,2}. But it doesn't give the answer s1s2s1.

W = CoxeterGroup(['A',4])
w = W.long_element()
t = w.coset_representative([1,2])
t.reduced_word()
lijr07 gravatar imagelijr07 ( 4 years ago )

You need to look at the other factor of the canonical factorisation

  sage: (~t*w).reduced_word()
  [1, 2, 1]
FrédéricC gravatar imageFrédéricC ( 4 years ago )

@FrédéricC, thank you very much for your very useful answer. Do you know how to find the unique element with maximal length in a double coset of a Weyl group: https://ask.sagemath.org/question/595... ?

lijr07 gravatar imagelijr07 ( 3 years ago )

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Asked: 4 years ago

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Last updated: Apr 13 '20