Is fully commutative for Coxeter group elements

[fchapoton]

📚 Description

This is adding a method "is_fully_commutative" to all Coxeter group elements.

On the way, introduce iterators over the sets of reduced words of a given element.

📝 Checklist

• The title is concise, informative, and self-explanatory.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

⌛ Dependencies

[fchapoton]

Maybe one could try instead to add a method "is_fully_commutative" to the BraidOrbit classes

[tscrim]

Can we add a test showing things works for a Coxeter group with a "be_careful" indexing set?

Also, personally, I think we should expunge the # optional markers until they are actually used (in particular, can be reasonably tested against).

[fchapoton]

Can we add a test showing things works for a Coxeter group with a "be_careful" indexing set?

Hello Travis, thanks for the idea. Would you please suggest a simple way to create such a group ?

(I prefer not to mess with these damn markers)

[tscrim]

There seems to be a bit of bug in the CoxeterMatrixGroup input ignoring the index set when passed something that can be interpreted as a Coxeter type.

Well, we can at least work around that by working with a Coxeter group for a very generic matrix:

sage: W = CoxeterGroup(CM, index_set=('a', Permutation([2,1]), -2), implementation="reflection")
sage: W.index_set()
('a', [2, 1], -2)

[fchapoton]

Thanks. Is my new doctest enough ?

EDIT: I see the index set is ignored there too.

[fchapoton]

should be better now

[tscrim]

Thank you. This is good.

[github-actions]

Documentation preview for this PR (built with commit dca133a; changes) is ready! 🎉