#15806: Integrable representations of (affine) Kac-Moody Lie Algebras
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Reporter: bump | Owner: bump
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.4
Component: combinatorics | Resolution:
Keywords: Kac-Moody | Merged in:
Authors: bump | Reviewers:
Report Upstream: N/A | Work issues: Due to a recursive
Branch: | caching scheme, it can crash for
public/combinat/integrable-15806 | hard computations. Obviously this
Dependencies: | should be fixed. Not all docstrings
| have doctests.
| Commit:
| 0aa8bb15e816f904bfff506453b7ef5721806a70
| Stopgaps:
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Comment (by bump):
Continuing from previous comment:
Proposition 12.6 describes the dominant maximal weights as follows. They
depend only on two things: the level `k` (currently implemented as
`self.level()` and a coset of the weight lattice in the root lattice
(called the class in KMPS). So different representations with the same
level and class will have the same number of dominant maximal weights, and
the same number of strings.
The weight lattice P has the classical weight lattice P0 of codimension
two. It is obtained from P0 by adjoining Lambda[0] and the nullroot delta.
The projection of the dominant maximal weights onto P0 has a simple
description. Let F be the fundamental alcove, and let `k F` be the
dilation of F by the level. So it is bounded by the innequalities
`alphacheck[i] >= 0` for i nonzero, and `alphacheck[0] <= k`, where
`alphacheck[0]` is the negative of the highest classical root.
Then the projections of the dominant maximal roots into P0 are just the
elements of `k F` that are in the right coset of the root lattice. If
these are known, then I don't think it will be difficult to recover the
dominant maximal roots. I believe that only the method `dominant_maximal`
needs to be reimplemented.
--
Ticket URL: <http://trac.sagemath.org/ticket/15806#comment:30>
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