#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:
Branch: | Commit:
public/combinat/integrable-15806 | 3a2328b9ee01824ca47991d70fa16d8e779c7600
Dependencies: | Stopgaps:
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Description changed by bump:
Old description:
> Integrable representations of Kac-Moody Lie algebras are parametrized by
> dominant weights. They satisfy a character formula which is a
> straightforward generalization of the Weyl character formula. See Kac,
> Infinite-dimensional Lie algebras, Chapter 10. The weight multiplicities
> may be computed using the Freudenthal multiplicity formula, and Kass,
> Moody, Patera and Slansky in their book Affine Lie algebras, weight
> multiplicities, and branching rules (1990) gave tables of these for
> affine Lie algebras. We may duplicate their tables using Sage, though at
> the moment only untwisted type A is implemented.
New description:
Integrable representations of Kac-Moody Lie algebras are parametrized by
dominant weights. They satisfy a character formula which is a
straightforward generalization of the Weyl character formula. See Kac,
Infinite-dimensional Lie algebras, Chapter 10. The weight multiplicities
may be computed using the Freudenthal multiplicity formula, and Kass,
Moody, Patera and Slansky in their book Affine Lie algebras, weight
multiplicities, and branching rules (1990) gave tables of these for affine
Lie algebras. We may duplicate their tables using Sage, though at the
moment only untwisted type A is implemented.
You can do the following with this code:
{{{
sage: Lambda =
RootSystem(['A',3,1]).weight_lattice(extended=true).fundamental_weights()
sage: IntegrableRepresentation(Lambda[1]+Lambda[2]+Lambda[3]).strings()
3*Lambda[2] - delta: 3 21 107 450 1638 5367 16194 45687 121876 310056
757056 1783324
2*Lambda[0] + Lambda[2]: 4 31 161 665 2380 7658 22721 63120 166085 417295
1007601 2349655
Lambda[1] + Lambda[2] + Lambda[3]: 1 10 60 274 1056 3601 11199 32354 88009
227555 563390 1343178
Lambda[0] + 2*Lambda[3]: 2 18 99 430 1593 5274 16005 45324 121200 308829
754884 1779570
Lambda[0] + 2*Lambda[1]: 2 18 99 430 1593 5274 16005 45324 121200 308829
754884 1779570
}}}
This creates the representation of `A_3^{(1)}` with highest weight
``Lambda[1]+Lambda[2]+Lambda[3]``. It then computes the five dominant
maximal weights and the corresponding weight strings. Every weight
multiplicity occurs in one of these strings, and by results of Kac and
Peterson, they are the Fourier coefficients of modular forms. The code is
reasonably fast and I've tested it up to rank 8.
--
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Ticket URL: <http://trac.sagemath.org/ticket/15806#comment:13>
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