#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: Travis Scrimshaw | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
public/combinat/fix_crystal_weight_lattice-18453|
b4a764743046481af4a6bb88ca353d607535d339
Dependencies: | Stopgaps:
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Changes (by {'newvalue': u'Travis Scrimshaw', 'oldvalue': ''}):
* commit: => b4a764743046481af4a6bb88ca353d607535d339
* branch: => public/combinat/fix_crystal_weight_lattice-18453
* author: => Travis Scrimshaw
Comment:
Okay, here's a rough draft of my proposal that mostly works. I would like
some feedback, in particular from Nicolas on the change of the behavior
and the reorganization of the code. There also seems to be a subtle issue
with subclassing and `@cached_method` (with my branch):
{{{
sage: P = RootSystem(['A',3,1]).weight_lattice()
sage: P.fundamental_weight('delta')
delta
sage: P.fundamental_weights()
Finite family {0: Lambda[0], 1: Lambda[1], 2: Lambda[2], 3: Lambda[3]}
}}}
Restart Sage and do
{{{
sage: P = RootSystem(['A',3,1]).weight_lattice()
sage: P.fundamental_weights()
Finite family {0: Lambda[0], 1: Lambda[1], 2: Lambda[2], 3: Lambda[3]}
sage: P.fundamental_weight('delta') # This is now the method of the base
class!!!
...
ValueError: delta is not in the index set
}}}
I have no idea about why this occurs, much less how to fix this.
However it does not actually fix the problem noted on this ticket as we
compute the weight by `Phi - Epsilon`. I've fixed this directly for LS
paths (to which we need to be much more careful with the category) and
rigged configurations. For Nakajima monomials, I'm not quite sure how to
fix it. I really don't want to special case the affine types. In Kang et
al., they construct B(\infty) by the path realization, and so I think they
consider it as a U',,q,,('''g''')-crystal (and so is in the non-extended
weight lattice).
(Additionally, the weight function isn't correct for the Kyoto path model
either, but that is a separate issue.)
----
New commits:
||[http://git.sagemath.org/sage.git/commit/?id=7663e78bdcbbb638818385b9d44efb6a05bdaf07
7663e78]||{{{Implement category of KR crystals and reorganizing (extended)
weight spaces.}}}||
||[http://git.sagemath.org/sage.git/commit/?id=6e0ba041ea9af34c913226cc81d47e87c5a0e58a
6e0ba04]||{{{Fixing my errors from the conversion.}}}||
||[http://git.sagemath.org/sage.git/commit/?id=b4a764743046481af4a6bb88ca353d607535d339
b4a7647]||{{{Fixing weight for LS paths and rigged configurations.}}}||
--
Ticket URL: <http://trac.sagemath.org/ticket/18453#comment:5>
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