On Fri, Oct 08, 2010 at 02:44:38PM -0700, Christian Stump wrote:
> Where I read about them, they were just called degrees (and codegrees,
> which will be important as well) of the reflection group, e.g. in
> Humphreys, but as well in more recent publications like in Drew
> Armstrongs thesis published in the memoirs of the AMS or the work by
> I. Gordon and S. Griffeth on non-well-generated complex reflection
> groups. So I would suggest to use just use this name.
Ok.
> In fact I plan to get Catalan numbers and q-Catalan numbers and (as
> soon as they are found somewhere out there ;-)) q,t-Catalan numbers
> available. They can be defined by using the degrees and codegrees,
> that's why I was looking for those...
Please discuss this with Mike Zabrocki, since he was doing much stuff
like that (but maybe only in MuPAD, and maybe only in type A).
> - CoxeterGroup are in fact finite Coxeter groups, and the few non-Weyl
> groups are not yet in sage.
Not quite:
sage: CoxeterGroup(["A",3,1])
Weyl Group of type ['A', 3, 1] (as a matrix group acting on the root space)
sage: CoxeterGroup(["H",3])
Permutation Group with generators
[(2,6)(3,18)(4,8)(5,9)(7,10)(11,12)(13,14)(17,21)(19,23)(20,24)(22,25)(26,27)(28,29),
(1,4)(2,17)(3,6)(5,7)(9,11)(10,12)(14,15)(16,19)(18,21)(20,22)(24,26)(25,27)(29,30),
(1,16)(2,5)(4,7)(6,9)(8,10)(11,13)(12,14)(17,20)(19,22)(21,24)(23,25)(26,28)(27,29)]
But yes, not all Coxeter groups are implemented, and the second one
above uses GAP3, which is not in plain Sage yet.
> - Cartan type (I prefer Cartan datum as well, as this is widely used)
> sounds to me like a classification type of a Cartan matrix (or more
> general a skew-symmetrizable matrix as the Cartan matrix of a Kac-
> Moody algebra).
Actually, a Sage CartanType is anything from which one can reconstruct
all the Cartan datum. It's not quite a classification type, since it
includes a specific choice of labeling of the nodes of the Dynkin diagram.
> - methods RootSystem.ambient_space and CartanType.AmbientSpace
The later is not a method but a nested class. Whether it should show
up during introspection is questionable. It could possibly be renamed
to _AmbientSpace. But then so should most other nested classes (like
the ParentMethods/... of categories.
> > - hardcode the data, type by type. Then, as suggested by Dan, this
> > should go in the files sage.combinat.root_system.type_???.py. But I
> > would put it in the data attached to the Cartan type, rather than
> > the ambient space:
>
> as mentioned above, this is data for any complex reflection group.
> They do *not* come with a root system or Cartan datum, so for them
> there exist nothing like the class ambient space.
> Cartan types
> generalize Weyl groups in a different direction than complex
> reflection groups. Those are classified as well, but the
> classification is the "Shephard-Todd" classification. So I would
> rather use ST classification for them.
I am speaking from the technical side here. For simplicity, data
around Coxeter groups and the like that depend on any sort of
classification would be best organized using the same file
organization as what's readily there. Admittedly the name of the
module "sage.combinat.root_system", and the type_??? might be
misnomers. We can change them later if really needed. Anyway that's
transparent from the user point of view.
> Are you talking about the connection to ranks in the root poset? You
> can get the degrees of a Weyl group by taking the integer partition
> \lamba where \lambda_i is given by the number of roots of height i.
> The degrees are then given by the parts of the transposed of \lambda.
> This gives a way to actually compute them for Weyl groups (when you
> loose crystallography, the roots have no integer heights anymore). But
> in general, there are no roots or anything in this direction
> available... Or are you talking about a different way to get the
> degrees?
That's indeed what I was thinking about. Though Jean Michel might have
generalizations. Jean?
> I will have a closer look at your references these days! I don't have
> the email of Jean Michel, so I cannot forward him this mail.
Oops, I fumbled my CC. This one should work!
Cheers,
Nicolas
--
Nicolas M. Thiéry "Isil" <[email protected]>
http://Nicolas.Thiery.name/
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