#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: | 9225af71b4dbcfa6c748816c452e9824dadc3651
public/crystal/18453 | Stopgaps:
Dependencies: |
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Comment (by tscrim):
I've fixed the rigged configurations
- `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
}}}
- 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.
- We have a problem with not distinguishing between elements of the
extended and non-extended affine weight lattices. In particular, this
causes an issue with creating the highest weight crystals with the same
weight but you accidentally first create it in the non-extended affine
weight lattice.
{{{
sage: LaE =
RootSystem(['A',2,1]).weight_space(extended=True).fundamental_weights()sage:
La = RootSystem(['A',2,1]).weight_space().fundamental_weights()
sage: B = crystals.LSPaths(La[0])
sage: B2 = crystals.LSPaths(LaE[0])
sage: B is B2
True
}}}
I had changed `crystals.RiggedConfigurations` with inadvertantly triggered
this. I don't think we want to enforce that the user must consider weights
in the extended affine weight lattice as it is a valid restriction from
U,,q,, to U,,q,,' (in particular, the Kyoto path model). IMO the solution
would be for `__classcall__`/`__classcall_private__` to pass the weight
lattice realization with the default to use the parent of the weight
given. I feel this gives the user added flexibility for where they want
the weights to be printed. Although part of me also thinks there should be
some kind of option which could be changed in the method-
input/classes/globally which specifies the weight lattice realization.
I will be on skype all day today if you want to talk.
--
Ticket URL: <http://trac.sagemath.org/ticket/18453#comment:14>
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