On Wednesday, January 4, 2017 at 9:00:19 PM UTC, Joseph Hundley wrote: > > Hi Travis and David, > > Sorry for posting and then disappearing for a bit. Thanks for your > answers. I will be very happy to have your help with the project to the > extent that you have time, and even just any quick suggestions and advice. > > My thinking was to do something that would cover arbitrary split connected > reductive algebraic groups using their classification by root data, and > associated presentations. For this one needs cases where the character > lattice is neither the root lattice nor the weight lattice. Maybe something > in between (like SO(n)) maybe something of higher rank (Like GL(n), > similitude groups, GSpin groups). > > It's not clear that this desire to do something general is that well > grounded in practical applications. In my old java project I implemented > only the five simply connected exceptional groups and mostly got by just > fine. Still, that was the idea. I noticed that reductive Kac-Moody groups > have a similar classification (and presentations) with root data. > > So, I think the implementation of root and weight lattices for a root > system is a good source of ideas but would be directly applicable only to > simply connected/adjoint semisimple groups. > > My thinking was that a based root datum can be easily passed as a pair of > matrices A,B. The rows of A represent the simple roots in terms of some > basis for the character lattice, and the rows of B represent the simple > coroots in terms of the dual basis of the cocharacter lattice. (One has to > work out the equivalence relation on matrix pairs that corresponds to > isomorphism of algebraic groups, but it's not too hard.) At that point one > has to choose bases for the two lattices in order to specify the object, so > they get an identification with ZZ^r essentially for free. Working out the > matrices for the two W-actions of simple reflections on ZZ^r from A and B > is pretty easy too. > > One thing I noticed is: it appears that the root and weight lattices are > implemented as ZZ-modules with a method .weyl_action, but not as > ZZ[W]-modules. Is there a reason for this choice? Naively, I thought "well, > what I have has the structure of a ZZ[W]-module so I guess I should > implement that structure." But you're both more experienced with Sage. > > I had been thinking to try to write something general which would take, as > input, a finitely presented group and a tuple of square integer matrices of > the same size. It would then check that mapping the generators in the > presentation to the matrices in order extends to a representation of the > group, i.e., to a ZZ[W]-module structure on ZZ^r for the relevant value of > r. Having done this checking it would then return that ZZ[W]-module. I > don't know whether that sounds like a good/bad approach to either of you. >
Such things are implemented in GAP (not exactly sure what is needed from the finitely presented group, it would certainly work in cases of presentations defining finite groups), so one just need to get hold of the corresponding GAP functionality. http://gap-system.org/Manuals/doc/ref/chap40.html#X7F348F497C813BE0 And, by the way, GAP also has functionality to construct irreducible complex representations of finite groups. Dima > > Best, > > Joe > > On Tuesday, December 13, 2016 at 4:34:23 PM UTC-5, Travis Scrimshaw wrote: >> >> Hey Joseph, >> There is currently a fair amount implemented with root systems: >> >> sage: Phi = RootSystem(['A',2]) >> sage: P = Phi.root_lattice() >> sage: Q = Phi.root_lattice() >> sage: P = Phi.weight_lattice() >> sage: al = Q.simple_roots() >> sage: al[1] >> alpha[1] >> sage: al[1].weyl_action([1,2,1]) >> -alpha[2] >> sage: al[1] + 4*al[2] >> alpha[1] + 4*alpha[2] >> sage: (al[1] + 4*al[2]).weyl_action([2,1]) >> 3*alpha[1] - alpha[2] >> sage: La = P.fundamental_weights() >> sage: La[1].weyl_action([1,2,1]) >> -Lambda[2] >> >> Will this work for your purposes? Let me know if you have any specific >> questions too. I'm happy to contribute how and when I can. >> >> Best, >> Travis >> >> >> On Tuesday, December 13, 2016 at 10:31:27 AM UTC-6, David Roe wrote: >>> >>> I'm interested in helping out with this project. Julian Rüth, Xavier >>> Caruso and I are working on p-adics on the Sage IRC channel ( >>> http://www.sagemath.org/help-irc.html) at 4pm EST (10pm CET) today. If >>> you want to log in and chat about character lattices for a bit I'd be happy >>> to share ideas. >>> >>> I'll try to respond to this e-mail with more specific suggestions about >>> existing structures in the next week or so, but before then I have to write >>> and grade finals.... >>> David >>> >>> On Tue, Dec 13, 2016 at 11:12 AM, Joseph Hundley <joseph....@gmail.com> >>> wrote: >>> >>>> I'm playing around with implementing Based Root Data, with an eye >>>> towards eventually implementing groups attached to them. >>>> So, I need to instantiate the character and cocharacter lattices; they >>>> need to be free Z-modules with an action of the relevant Weyl group. In >>>> other words they are Z[W]-modules. Any suggestions on relevant existing >>>> structures would be welcome. >>>> >>>> -- >>>> You received this message because you are subscribed to the Google >>>> Groups "sage-devel" group. >>>> To unsubscribe from this group and stop receiving emails from it, send >>>> an email to sage-devel+...@googlegroups.com. >>>> To post to this group, send email to sage-...@googlegroups.com. >>>> Visit this group at https://groups.google.com/group/sage-devel. >>>> For more options, visit https://groups.google.com/d/optout. >>>> >>> >>> -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
