#15361: Branching Rules for Exceptional Groups
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Reporter: bump | Owner: bump
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.1
Component: combinatorics | Resolution:
Keywords: | Merged in:
Authors: bump | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
public/combinat/15361-branching- | 5fbd7f6d674dfd53eb70cba009fa6294cac68db1
rules | Stopgaps:
Dependencies: |
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Description changed by bump:
Old description:
> Branching rules for Lie groups are mostly already implemented in
> {{{weyl_characters}}}. That is, if G is a Lie group and H a subgroup
> (maximal without loss of generality) we can compute the branching rule
> from {{{G => H}}} in most cases, always if G is of classical type, and
> sometimes if G is an exceptional group.
>
> Before the patch, the following rules are not implemented.
>
> {{{
> E6 => C4 , A2 , G2 , A2xG2
> E7 => A2 , A1 , A1 , A1xF4 , G2xC3 , A1xG2 , A1xA1
> E8 => G2xF4 , C2 , A1xA2 , A1 , A1 , A1
> F4 => A1 , A1xG2
> G2 => A1
> }}}
>
> With the patch, ALL of these are now implemented in Sage. After the
> patch, every
> branching rule in the tables of McKay and Patera (Tables of dimensions,
> indices and branching rules for representations of simple Lie algebras)
> is available in Sage!
>
> The embeddings are described in the thematic tutorial. I've posted a copy
> of the patched tutorial here:
>
> http://sporadic.stanford.edu/thematic_tutorials/lie/branching_rules.html
>
> The patch makes a class BranchingRule for branching rules. Notable
> methods are a multiplication corresponding to composition, and a
> {{{describe()}}} method for branching rules which shows how simple roots
> and the affine root restrict. The multiplication gives a better method of
> concatenating branching rules. A projection method for composite types is
> given. The goals set out in Comment 6 are all achieved. The thematic
> tutorial is revised.
New description:
Branching rules for Lie groups are mostly already implemented in
{{{weyl_characters}}}. That is, if G is a Lie group and H a subgroup
(maximal without loss of generality) we can compute the branching rule
from {{{G => H}}} in most cases, always if G is of classical type, and
sometimes if G is an exceptional group.
Before the patch, the following rules are not implemented.
{{{
E6 => C4 , A2 , G2 , A2xG2
E7 => A2 , A1 , A1 , A1xF4 , G2xC3 , A1xG2 , A1xA1
E8 => G2xF4 , C2 , A1xA2 , A1 , A1 , A1
F4 => A1 , A1xG2
G2 => A1
}}}
With the patch, ALL of these are now implemented in Sage. After the patch,
every
branching rule in the tables of McKay and Patera (Tables of dimensions,
indices and branching rules for representations of simple Lie algebras) is
available in Sage!
Here is a file that constructs the branching rule for every maximal
subgroup of every simple Lie group of rank less than or equal to 8. This
includes every case considered by McKay and Patera, and every exceptional
group.
http://sporadic.stanford.edu/bump/branch-table.sage
The embeddings are described in the thematic tutorial. I've posted a copy
of the patched tutorial here:
http://sporadic.stanford.edu/thematic_tutorials/lie/branching_rules.html
The patch makes a class BranchingRule for branching rules. Notable methods
are a multiplication corresponding to composition, and a {{{describe()}}}
method for branching rules which shows how simple roots and the affine
root restrict. The multiplication gives a better method of concatenating
branching rules. A projection method for composite types is given. The
goals set out in Comment 6 are all achieved. The thematic tutorial is
revised.
--
--
Ticket URL: <http://trac.sagemath.org/ticket/15361#comment:53>
Sage <http://www.sagemath.org>
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and MATLAB
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