#18453: Infinite affine crystals should use extended weight lattice
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Reporter: bump | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-6.8
Component: combinatorics | Resolution:
Keywords: crystals, days65 | Merged in:
Authors: Ben Salisbury, | Reviewers:
Anne Schilling, Travis Scrimshaw | Work issues:
Report Upstream: N/A | Commit:
Branch: | 3b3c0b2d15289a8c4a5c5c25ad2984114a856916
public/crystal/18453 | Stopgaps:
Dependencies: |
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Comment (by tscrim):
Replying to [comment:20 aschilling]:
> > - `CrystalOfAlcovePaths` fails outright:
> No, it does not. You did not send how you made the weight La. I assume
you used the weight_lattice and not weight_space (which you have to for
this model!!)
Ah, we do need the `weight_space`, so we should probably make it error out
similar for the LS paths if we aren't in the weight space. Also it does
work using the extended weight lattice:
{{{
sage: La =
RootSystem(['A',2,1]).weight_space(extended=True).fundamental_weights()
sage: C = crystals.AlcovePaths(La[0])
sage: C.module_generators[0].f_string([0,1])
((alpha[0], 0), (alpha[0] + alpha[1], 0))
sage: _.weight()
-Lambda[1] + 2*Lambda[2] - delta
}}}
> > - We should also explicitly implement a `weight_lattice_realization`
for `DirectSumOfCrystals` similar to what we did for tensor products.
> > - We have to decide what we want to do with `KyotoPathModel` and if we
want to consider it as a U,,q,,'-crystal or a U,,q,,-crystal. I think the
former is what we should do considering it is a tensor product of
U,,q,,'-crystals. In any case, we will probably have to do something
special for this.
>
> Mathematically speaking, the Kyoto path model is a model in the category
of highest weight affine crystals. It is a U_q(g)-crystal not a U_q'(g)
crystal. Really what one does it take tensor products `B^{r,s} \otimes
u_\Lambda` with `\Lambda` of level `s`.
I agree that it is in highest weight affine crystals, but we don't have a
way to get `\delta` from the elements. The embedding `B(\lambda) \to
B^{r,s} \otimes B(\mu)` has to be as U_q'(g)-crystals as `B^{r,s}` must be
a U_q'(g)-crystal as otherwise it breaks the condition for the weight
function. Also if `\Lambda` is of level `s`, then `\Lambda + k\delta` is
of level `s` for all `k` since `<c, \delta> = 0`. I still don't quite see
how we could figure out what the coefficient of `\delta` is for an
arbitrary element in the Kyoto path model.
Replying to [comment:21 bsalisbury1]:
> It seems the original error in the monomial crystals code is due to an
error in the literature about how the weight function is defined for
monomial crystals.
> ...
> In none of these references is there a way to get delta to appear as a
weight. Perhaps I've missed a reference that does include this
information, but I'm thinking about the problem now and planning to
discuss it with Peter Tingley, so hopefully we will have a resolution
soon.
I had poked around on this too when I first looked into this ticket and
couldn't find anything (I also looked in ''Level 0 monomial crystals'' by
Hernandez and Kashiwara). Expressing the monomial crystals in the `A`'s
isn't a problem since we can just pull the simple roots and `\alpha_0` is
what contributes to `\delta`. We can always brute-force this computation
for highest weight crystals by taking a path to the highest weight and
computing the weight from that, but that is going to be ''really'' slow
(but it will be correct). I think the best solution is to follow Hernandez
and Kashiwara and add a weight attribute to each element of the crystal,
which is easy enough to compute on each application of `e` and `f`.
Also in a similar vein, I don't think we can compute `\delta` from doing
`Epsilon - Phi` as `Epsilon = \sum_i \epsilon_i \Lambda_i` and similar for
`\Phi` (which comes down to the fact that `<h_i, \delta> = 0`, where `h_i`
is a simple coroot). I think we need to change this, or at least put a
stopgap in affine type and the WLR is the extended weight lattice.
--
Ticket URL: <http://trac.sagemath.org/ticket/18453#comment:23>
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