Salut Nicolas, > As far as I know, this is not yet implemented. And it's definitely in > the TODO list! Any good name for this method? I am using > invariant_degrees below, in waiting for something better.
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. 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... > > There is a way to get them by W.invariant_generators(), but this > > took for S_4 already half a minute. As many of you know, those > > degrees are very important data for the group, as many things can be > > computed using those. > > YES. > > > While looking for those degrees (maybe I just missed them in some > > natural place...), I saw that there are many things in this area > > looking unorganized and too complicated. > > Please elaborate! some random things I saw: - CoxeterGroup are in fact finite Coxeter groups, and the few non-Weyl groups are not yet in sage. - 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). - methods RootSystem.ambient_space and CartanType.AmbientSpace let me have a closer look these days, and I will better organize my thoughts in this direction. > Let me use the occasion to > advertisehttp://trac.sagemath.org/sage_trac/wiki/SageCombinatChevieStatusReport > which points to some of the relevant tickets. We should add more > tickets for whatever needs improvement. I will look at those as well. > > I think of starting to get first basic things of general Coxeter > > groups and of complex reflection groups into sage (of cause without > > using gap - now as the universal cyclotomic field is in sage) and as > > well to organize things in this direction... What do you think? > > +1! Actually, I have an ongoing research project involving complex > reflexion group, so I might be your first beta tester :-) Very good! > For the implementation of the degrees of the fundamental invariants, > there are two options: > > - 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. > - Jean Michel (in CC) mentioned a nice trick of his to compute those > in a type free manner from some statistics on the heights of the > roots, or something like this. I don't remember if this was for > Weyl groups or general reflections groups though. Jean Michel, > could you elaborate? 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? 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. Thanks, Christian -- You received this message because you are subscribed to the Google Groups "sage-combinat-devel" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/sage-combinat-devel?hl=en.
