#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 | Merged in:
Authors: Ben Salisbury, | Reviewers:
Anne Schilling, Travis Scrimshaw | Work issues:
Report Upstream: N/A | Commit:
Branch: | 7963ea6a64dc4bd1d623288fbb6702e11beae0dc
public/crystal/18453 | Stopgaps:
Dependencies: |
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Comment (by aschilling):
> - `CrystalOfAlcovePaths` fails outright:
> {{{
> sage: C = crystals.AlcovePaths(La[0])
> sage: C
> Highest weight crystal of alcove paths of type ['A', 1, 1] and weight
Lambda[0]
> sage: C.module_generators[0].f_string([0,1])
> ((alpha[0], 0), (2*alpha[0] + alpha[1], 1))
> sage: _.weight()
> TypeError
> }}}
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!!)
{{{
sage: R = RootSystem(['A',1,1])
sage: P = R.weight_space()
sage: La = P.basis()
sage: C = crystals.AlcovePaths(La[0])
sage: b = C.module_generators[0].f(0)
sage: b
((alpha[0], 0),)
sage: b.weight()
-Lambda[0] + 2*Lambda[1]
}}}
> - 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`.
--
Ticket URL: <http://trac.sagemath.org/ticket/18453#comment:20>
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