Automate Branching Rules for IntegrableRepresentations

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Currently branching rules for IntegrableRepresentations of untwisted affine Lie algebras are implemented. But the process requires the user to create a dictionary between the nodes of two Dynkin diagrams, a process that can be automated.

Depends on #18767

CC: @sagetrac-sage-combinat

Component: combinatorics

Author: Daniel Bump

Branch/Commit: public/combinat/branching-15398 @ 1706a8d

Issue created by migration from https://trac.sagemath.org/ticket/18398

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New commits:

3779d1a

get_branching_rule method

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Branch: public/combinat/branching-15398

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Commit: 3779d1a

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Branch pushed to git repo; I updated commit sha1. New commits:

186389c

automated branching rules now working

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 3779d1a to 186389c

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Changed commit from 186389c to 1ee0267

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Branch pushed to git repo; I updated commit sha1. New commits:

1ee0267

doc revision

[tscrim]

comment:5

I think it would be better if we implement a method on the various CartanType classes which does the respective branching rules. In #15463, I go part of the way there by implementing a subtype method, but I think we should take that idea and expand on it here, and then use subtype to do the branching rule here. Does this sound like a good approach or do you need something more direct here (FYI - I haven't looked too closely at this patch yet)?

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Work Issues: #18767 should be merged first since it is more important

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comment:6

This patch contains a lot of case-by-case encoding of work that can also be done by hand by the user when the user needs the branching rules (explained in the reference manual). It saves the reader having to line up the Dynkin diagrams.

So although it is convenient not to have to do this, the reader who needs a branching rule can do so without this patch. This patch is therefore less important than #18767 which adds real new functionality. After #18767 this patch should be generalized to include the twisted types. So I think it is better if #18767 (which adds real new functionality) is merged first. After that, this patch should be extended to include the twisted cases.

In #15463, I go part of the way there by implementing a subtype method, but I think we should take that idea and expand on it here, and then use subtype to do the branching rule here.

Following a suggestion of Volker Braun, branching rules for finite dimensional Lie algebras were implemented in the categorical framework in the Weyl character ring.

sage: R1=branching_rule("E7","A7","extended")
sage: type(R1)
<class 'sage.combinat.root_system.branching_rules.BranchingRule'>
sage: R2=branching_rule("A7","A3xA3","levi")
sage: R1*R2
composite branching rule E7 => (extended) A7 => (levi) A3xA3

One could try to extend this to the affine case but here we are only dealing with Levi branching rules. (The Levi subalgebras are finite-dimensional.) This is less ambitious but still useful.

[tscrim]

comment:7

We could probably benefit from using the subtype method in #17798 (and could potentially need improvements) so it requires less hard-coded branching rules.

[tscrim]

Dependencies: #18767

[tscrim]

Changed work issues from #18767 should be merged first since it is more important to none

[cheuberg]

comment:8

multiple doctests fail, cf. patchbot

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Changed commit from 1ee0267 to 1706a8d

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Branch pushed to git repo; I updated commit sha1. New commits:

79963f8	Merge branch 'public/combinat/branching-15398' in 7.6.b6
1706a8d	trac 18398 python3 print