#15508: Implement Fock space
-------------------------------------+-------------------------------------
Reporter: tscrim | Owner: sage-combinat
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-7.3
Component: algebra | Resolution:
Keywords: Fock space | Merged in:
quantum group representations |
Authors: Travis Scrimshaw | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
public/modules/fock_space | e0ea20cf70e5df0525615683ec82c127b62b3acd
Dependencies: #15289 #15525 | Stopgaps:
#15621 |
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Changes (by tscrim):
* milestone: sage-6.4 => sage-7.3
Comment:
Replying to [comment:36 andrew.mathas]:
> Replying to [comment:35 tscrim]:
> >Whoops, forgot `rank = n - 1`.
>
> In the Kac-Moody world I think that `rank=n` is correct. In the example
above, I thought that only `.f(0)`, `.f(1)` and `.f(2)` should work, but
not `.f(4)`, `.f(5)` etc.
Ah, I thought you were referring to the (usual matrix) rank of the Cartan
matrix. I think it is better to have the `rank = n` (i.e., the Kac-Moody
version), and I will make sure this is well-documented.
> > One of the main reasons why I want to keep them separate is because
the Fock space can be larger since it has non-regular partitions as basis
elements. The current version of the highest weight representation is only
considered as Uq(sln)-repr instead of a Uq(gln)-repr. At some point I will
find some time to implement the Uq(gln) case (sorry I haven't Anne!), and
at which point, I would have that be a basis for the Fock space. I also am
not sure how the natural basis would act under e/f, in fact, I am not sure
the natural basis makes sense in the Uq(sln)-representation... So I think
they should be separate.
>
> The full Fock space is a a `U_q(\widehat{sl}_n)`-module, it's just not
irreducible, and the action of `e` and `f` is what you have implemented.
Yes, right. I either ended up misremembering things or was thinking in
terms of irreducibility. Thinking about it more (and correctly now), it
would be good to have the approximation and upper global crystal bases as
bases for the full Fock space, with appropriate error messages for the
cases where they are not implemented. I am thinking we should keep the
irreducible part as well so we could make more of a connection with the
actual crystals later on in the form of explicitly computing the Kashiwara
operators. Do you agree?
> If you are able to implement the full `U_q(\widehat{gl}_n)` action that
would be awesome, although I think this will be hard. I can implement the
canonical basis for `U_q(\mathfrak{gl}_\infty)`, but I would need to
think/read about how to do the full `U_q(\mathfrak{gl}_\infty)`-action.
I think this is doable, but it would require quite a bit of work (and
whose knows how computationally feasible it would be). Definitely things
for follow-up tickets.
--
Ticket URL: <https://trac.sagemath.org/ticket/15508#comment:37>
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