Hey Andrew and Nicolas, I've probably mentioned this before, but I think we should figure out how we want to handle categories for representations in general. I would like this for Lie algebras (equiv. universal enveloping algebras), quantum groups, and general groups (equiv. group algebras). Something which would be benefit from having a category for Hecke algebra representations would be methods for constructing the Grothendieck group, a list of simples (up to isomorphism), and a canonical place to look for constructing examples (via ``an_element()``).
So my thoughts (if you've heard this before, feel free to skip) on this would be to construct a category parameterized by a Hecke algebra ``H`` which is a subcategory of the category of modules over ``H.base_ring()``. For general representation categories, we'd have an abstract method ``representation_action()`` (the name can change, I didn't think too hard about this) that the user must implement and which we use throughout the category and with some default implementation of ``get_action()`` which calls ``representation_action()`` when passed an element of ``H``. In this case specifically, we could define a generic ``representation_action()`` which converts an element to the T_i basis and calls ``T_action(i)`` (again, probably a better name out there). Although for now we can just have a base class for all Hecke algebra representations, and there is not a clear and obvious gain to me at present from defining such a category. Best, Travis On Friday, November 21, 2014 4:16:13 PM UTC-8, Andrew wrote: > > > > On Saturday, 22 November 2014 10:24:43 UTC+11, Nicolas M. Thiery wrote: >> >> Hi Andrew, >> Hmm, I haven't taken the time to think with a pen and paper, but I >> don't guarantee that the what the super call do is as straightforward >> as it ought to, given that the instance of A being initialized will >> end up being in a subclass A_with_category, and similarly for the >> others. Does it work if instead you call explicitly A.__init__ in >> B.__init__, and so on? >> > > Hi Nicolas, > > I fixed my problem so the code is working now. It is possible that there > are some issues with the super calls but as far as I can tell it is working > as I expect. > > One question that I should ask you, however, is the following. I have a > suspicion that I should define a category for the Hecke algebra > representations, however, I don't know what the benefits are of doing this > or how to do it. Certainly the code seems to work without this (not that I > have added extensive tests yet or properly implemented the mathematics), > but if this is necessary, or desirable, I would like to understand what > this gives me over and above having a dedicated class for Hecke algebra > representations. > > Andrew > > > -- You received this message because you are subscribed to the Google Groups "sage-combinat-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-combinat-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-combinat-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-combinat-devel. For more options, visit https://groups.google.com/d/optout.
