Dear Martin, > On 19 Dec 2017, at 09:48, Martin Rubey <martin.ru...@tuwien.ac.at> wrote: > > "Nicolas M. Thiery" <nicolas.thi...@u-psud.fr> writes: > >> On Mon, Dec 18, 2017 at 01:57:04PM +0100, Martin Rubey wrote: >>> I admit I am not completely sure right now: can we associate a conjugacy >>> class of subgroups of S_n, isomorphic to the Weyl group, in a sensible >>> way to a finite Cartan type? >> >> The permutation action on the roots would be a rather natural choice. >> This could be extended to affine Weyl groups using affine >> permutations.
Note that even different (finite) Cartan types can lead to isomorphic Weyl groups, unless you restrict to irreducible ones. And if you also permit infinite Weyl groups, then even for the irreducible ones, there are pairs of different irreducible types which have isomorphic Weyl groups; but from all you write, I guess you are only interested in finite groups. But of course they induce distinct "permutation isomorphism classes", to use the nomenclature Thomas' described. Which to me suggests that what Thomas describes really is a better fit for what you want to do. Regarding Nicolas' suggestion: While it is a "natural" choice, it is not necessarily a "canonical" choice. At the very least, you'd have to define how to label the roots "canonically". But OK, that can be done, perhaps you are even doing it already. But now to identify permutation group with your "canonical form", you need to solve a subgroup conjugacy problem, which can be hard. Moreover, it is not clear to me how to, given a permutation group, you'd compute its "canonical form" intrinsically (i.e. other than going through your whole database, then trying to compute a permutation isomorphism with each "canonical form" in the database, which of course is kinda against the whole point in defining a canonical form in the first place.) So I'd be very hesitant to call this a "canonical form", because to me, the name suggests that it should be possible to compute it intrinsically (i.e. w/o referring to a big table of "canonical forms"), and somewhat efficiently (as otherwise, there is little use). Of course that's just my point of view, you can disagree, but then I wonder what the point behind having those "canonical forms" really is? > > Indeed! This leaves the question whether there we can compute a > canonical form for permutation groups up to conjugacy. I just noticed > that this is a mathoverflow question, too: > > https://mathoverflow.net/q/90107/3032 > > Unfortunately, as of today it doesn't have a straight answer, but maybe > things have changed? No. This is still an incredibly tough problem. Cheers, Max _______________________________________________ Forum mailing list Forum@gap-system.org https://mail.gap-system.org/mailman/listinfo/forum
