#18453: Infinite affine crystals should use extended weight lattice
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Reporter: bump | Owner:
Type: defect | Status: needs_review
Priority: major | Milestone: sage-6.8
Component: combinatorics | Resolution:
Keywords: crystals, days65 | Merged in:
Authors: Ben Salisbury, | Reviewers: Dan Bump
Anne Schilling, Travis Scrimshaw | Work issues:
Report Upstream: N/A | Commit:
Branch: | f4955c5f50d4f34510176acff8956f49d3b8702a
public/crystal/18453 | Stopgaps:
Dependencies: |
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Comment (by bump):
From the thematic tutorial at
http://doc.sagemath.org/html/en/thematic_tutorials/lie/affine_hw_crystals.html
we have:
{{{
sage: sage: La =
RootSystem(['C',3,1]).weight_space().fundamental_weights()
sage: sage: LS = crystals.LSPaths(2*La[1]+La[2])
sage: sage: SL = LS.subcrystal(max_depth=3)
sage: sage: GL = LS.digraph(subset=SL)
sage: sage: La =
RootSystem(['C',3,1]).weight_lattice().fundamental_weights()
sage: sage: M = crystals.NakajimaMonomials(['C',3,1], 2*La[1]+La[2])
sage: sage: SM = M.subcrystal(max_depth=3)
sage: sage: GM = M.digraph(subset=SM)
sage: sage: GL.is_isomorphic(GM, edge_labels=True)
True
}}}
Why do we use weight_space for one and weight_lattice for the other?
(Maybe the thematic tutorial should explain.)
After this:
{{{
sage: LS.weight_lattice_realization().is_extended()
False
sage: M.weight_lattice_realization().is_extended()
True
}}}
I think the point of the example in the tutorial is that the two crystals
are isomorphic, but one uses the extended lattice and the other does not.
--
Ticket URL: <http://trac.sagemath.org/ticket/18453#comment:44>
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