As we now discussed in private, the translation factors for type A_{2n}^{(2)}
with the specifications as given in sage should be 1,1,...,1,1/2.Anne Nicolas M. Thiery wrote:
Hi Anne! On Wed, May 26, 2010 at 06:02:04PM -0700, Anne Schilling wrote:There remains to double check the translation factors for type A_2n^2;For A_{2n}^{(2)}, c_\alpha = 1,1,2 for (alpha,alpha)=1,2,4 respectively. This seems to be opposite of what is there right now sage: C=CartanType(['A',4,2]) sage: C.dynkin_diagram() O=<=O=<=O 0 1 2 BC2~ sage: R = RootSystem(C).weight_lattice() sage: c = R.cartan_type().translation_factors() sage: c Finite family {0: 2, 1: 1, 2: 1} But then, I think you have dualized everything, since I would also expect c_alpha = 1 for type C_3^{(1)} and you get sage: C=CartanType(['C',3,1]) sage: R = RootSystem(C).weight_lattice() sage: c = R.cartan_type().translation_factors() sage: c Finite family {0: 1, 1: 2, 2: 2, 3: 1} If you have dualized, then it seems to be correct.Thanks for investigating! I should have made a precise request though. What I would like to be 100% sure about is that this method implements its specification, which is: Returns the translation factors for ``self``. Those are the smallest factors `t_i` such that the translation by `t_i alpha_i` sends the fundamental polygon to another polygon in the alcove picture. Note that this spec is unambiguous about dualizing or not. Also, whether and when those factors t_i actually match with the usual c_{\alpha_i} or the ccheck_{\alpha_i}, or none of them, is certainly a useful information, well worth mentioning in the documentation, but this is not the main point. I pointed to the MuPAD-Combinat code because there we had apparently made a special case for A_2n^2, and the translation factors involved 1/2 coeffs. A distinct question is whether we want to also provide methods for the c_{\alpha_i} and ccheck_{\alpha_i}. And also whether it would be more natural / practical if translation_factors would be about translations by simple coroots instead; in that case, the specs, code, and use of this method should be updated all at once. Cheers, Nicolas PS: btw, tell me if you know / find a *conceptual* definition for the c_{\alpha_i} / ccheck_{\alpha_i}. So far, I only remember seeing them defined by a quite meaningless formula (max(1,...)).
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