Hello, On Sat, Oct 09, 2010 at 12:13:45AM +0200, Nicolas M. Thiery wrote: > 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. ... > That's indeed what I was thinking about. Though Jean Michel might have > generalizations. Jean?
I received 3 messages, I do not know to which reply -- I choose this one. All you want is implemented in Chevie, which is accessible via the GAP3 interface, so that's one solution. Note that Gordon and Griffeth that you quote use Chevie. Invariants, Degrees, Codegrees are implemented in Chevie in 2 ways: -stored values type by type which are returned using the type-recognition routines (recognizing the type of a finite Coxeter group is fairly trivial, but there is a lot of code in Chevie to do type-recognition for complex reflection groups). -general algorithms which compute the degrees. The codegrees of finite Coxeter groups are just the degrees minus 2, but there is no general algorithms for the codegrees or invariants for complex groups. One has to use the type-by-type data via type recognition. For the degrees: -The list of exponents (degrees minus one) of a finite Coxeter group is just the dual partition of the partition of the positive roots by height. -in BonnaféLlehrerMmichel Twisted invariant theory for reflection groups Nagoya Math. J. 182 (2006) 135--170 I give on page 164 a product formula (in the context of reflection cosets, but one can just set $\gamma=1$ for the trivial coset) which can be used -- and has been programmed in Chevie, in the routine PermRootOps.ReflectionDegrees -- to compute the degrees starting from the reflection representation of any finite complex reflection group. It uses itself the list of eigenvalues of elements in the reflection representation, which just requires to know the character of the reflection representation and its exterior powers (see PermRootOps.ReflectionEigenvalues). -------------------- I recommend in general for such questions, if you want to re-implement all I have done, that you just read the code of Chevie. I hope it is readable, and if not please tell me, I will add documentation and give you any explanations needed. Best regards, ------------------------------------------------------------------------ Jean MICHEL, Equipe des groupes finis, Institut de Mathematiques UMR7586 Bureau 9D17 tel.(33)157279144, 175, rue du Chevaleret 75013 Paris -- 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.
