Coxeter groups: more Bruhat and weak order features

[nthiery]

New methods:

• bruhat_le (code inspired from code by Daniel Bump)
• weak_le
• bruhat_poset (finite Coxeter groups)
• weak_poset (finite Coxeter groups)

Improved doctests for related methods

(Trivially) depends on #7842

[bump]

This patch implements the bruhat_order as a cached method and is badly needed.

With Sage 4.3, this raises an exception at the test in coxeter_groups.py, line 1010.

The definition of Q could be rewritten:

W = WeylGroup("B3")
sage: fcn = lambda x,y : x.bruhat_le(y)
sage: Q=Poset((W.list(),fcn))

Maybe the ?!? should be removed from the # long time directive a couple of lines later since it is unclear what it means.

[nthiery]

Replying to bump:

This patch implements the bruhat_order as a cached method and is badly needed. With Sage 4.3, this raises an exception at the test in coxeter_groups.py, line 1010.

Oops, I forgot that this depended on another patch; now #7842. As you mention, this dependency is trivial though, so we can also work around it if #7842 is not merged instantly.

Maybe the ?!? should be removed from the # long time directive a couple of lines later since it is unclear what it means.

Fixed, and updated the # long time around that line. I was just surprised by how much time this was taking. We need more Weyl group optimizations!

[nthiery]

Updated timings w.r.t. #7754 which is already in Sage

[bump]

By now this code is tested a lot, at least for finite Weyl groups, and the previous reviewer comments were addressed. I am changing the status to positive review.