Hey Andrew,
Am I right in thinking that sage does not (yet) know about the left, right
> and two-sided Kazhdan-Lusztig cells of Coxeter groups? As sage can compute
> Kazhdan-Lusztig polynomials I assumed that it knew about cells as well, but
> it doesn't seem to. Please tell me if I am missing something.
>
AFAIK, there's no code in Sage or ticket about cells. +1 to adding them.
>
> Incidentally, I also noticed that we have some identity issues -- that I
> am sure are well known:
>
> sage: WeylGroup("A5")
> Weyl Group of type ['A', 5] (as a matrix group acting on the ambient space
> )
> sage: CoxeterGroup("A5")
> Coxeter group over Universal Cyclotomic Field with Coxeter matrix:
> [1 3 2 2 2]
> [3 1 3 2 2]
> [2 3 1 3 2]
> [2 2 3 1 3]
> [2 2 2 3 1]
>
> I am guessing that one of these is just wrapping code from chevie, but I
> haven't checked. Is there any reason not to amalgamate these two classes?
> Currently the CoxeterGroup code seems marginally faster but perhaps the
> WeylGroup classes have more functionality:
>
I implemented the current CoxeterGroup and it uses Sage's (hence gap's)
matrix group code. However the WeylGroup does have more information
(features) since it is considered acting on the root/weight lattice, such
as reflection_to_root on elements. Combining these classes might be taken
care of by http://trac.sagemath.org/ticket/15703.
>
> sage: WeylGroup("A5").cardinality()
> 720
> sage: CoxeterGroup("A5").cardinality()
> ---------------------------------------------------------------------------
> NotImplementedError Traceback (most recent call last
> )
> <ipython-input-6-42496321af01> in <module>()
> ----> 1 CoxeterGroup("A5").cardinality()
>
> /usr/local/src/sage/local/lib/python2.7/site-packages/sage/categories/
> sets_cat.pyc in cardinality(self)
> 1370 NotImplementedError: unknown cardinality
> 1371 """
> -> 1372 raise NotImplementedError("unknown cardinality")
> 1373
> 1374 # Functorial constructions
>
> NotImplementedError: unknown cardinality
>
> There's currently http://trac.sagemath.org/ticket/16630 which should fix
some (all?) of these issues (and I will get to that after next week).
Best,
Travis
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