#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 bsalisbury1):
Replying to [comment:23 tscrim]:
> 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`.
I tried doing this using the path to the highest weight vector, but it
causes a loop in Sage because of the way the crystal operators are defined
in the monomial crystals model. In particular, we need phi to compute the
action of the Kashiwara operators, and phi depends on the weight in the
B(infinity) model.
--
Ticket URL: <http://trac.sagemath.org/ticket/18453#comment:24>
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