Opened 5 years ago
Closed 5 years ago
#15361 closed enhancement (fixed)
Branching Rules for Exceptional Groups
Reported by:  bump  Owned by:  bump 

Priority:  major  Milestone:  sage6.1 
Component:  combinatorics  Keywords:  
Cc:  bump, sagecombinat, vbraun  Merged in:  
Authors:  Daniel Bump  Reviewers:  Volker Braun 
Report Upstream:  N/A  Work issues:  
Branch:  public/combinat/15361branchingrules (Commits)  Commit:  d3e1db5eef471eb9eb935804c011085e6aa0ac49 
Dependencies:  Stopgaps: 
Description (last modified by )
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/branchtable.sage
The embeddings are described in the thematic tutorial. I've posted a copy of the patched tutorial and reference manual on sporadic.stanford.edu. The relevant sections are here:
http://sporadic.stanford.edu/thematic_tutorials/lie/branching_rules.html
http://sporadic.stanford.edu/reference/combinat/sage/combinat/root_system/weyl_characters.html
http://sporadic.stanford.edu/reference/combinat/sage/combinat/root_system/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.
Since weyl_characters.py was getting huge, I split it, moving the branching rule material into a new file, branching_rules.py.
Change History (76)
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35f73af  implemented E6>G2xA2 branching rule and corrected wrong timing info on long tests 
comment:9 Changed 5 years ago by
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fa2255a  implemented multiplication of branching rules 
ecadb7a  Merge branch 'public/combinat/15361branchingrules' of trac.sagemath.org:sage into 15361branchingrules 
dcd5477  updated timing information and implemented E6>G2xA2 branching rule 
36e6b24  make CartanType? idempotent in get_branching_rule 
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3372f1e  revision of branching rules, including composite rules 
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9b2121d  F4 => G2xA1 branching rule implemented 
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4a4bac8  corrections and revisions to existing branching rules and tests 
37ad3a2  Use compact representation for Cartan type in WeylCharacterRing? repr method 
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e9ccead  further cleanup and corrections 
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efb2a92  E8>E7,E7>E6 rules gave correct answers but did not map weight lattice into weight lattice, hence were broke for composition. This is fixed. 
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3ed9056  projection branching for reducible types 
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e656f78  improved repr for class BranchingRule? 
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c1e9107  revised G2=>A2 rule, added describe() method to class BranchingRule? 
a8cbda3  revisions to repr method of BranchingRule? 
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6b31a25  describe method of BranchingRule? 
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8de2532  work on describe method of BranchingRule? 
comment:23 Changed 5 years ago by
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5e9a05d  revision of describe method, E6 => G2xA2 branching 
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b4a5cce  revision of thematic tutorials regarding branching rules

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2d8bd61  E7 => G2xC3 branching rule implemented.

comment:28 Changed 5 years ago by
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comment:29 Changed 5 years ago by
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95d9098  doc revision regarding orthogonal branching rules

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7e38786  thematic tutorial revision

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ac6cfe3  several rules implemented, including A1 embeddings into exceptional groups.

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4fffc86  added branching rules E6 => G2,A2 per Testerman (1989)

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d8c75d9  thematic tutorial: minor cleanup

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7185c7d  thematic tutorial: minor cleanup

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e68046a  E7=>A1xA1 branching rule implemented

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f0474ad  E7 => A2 branching rule implemented

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2afcfe7  E8 => B2 rule implemented. Exceptional branching rules are now complete

comment:50 Changed 5 years ago by
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comment:51 Changed 5 years ago by
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e45ae6f  tutorial fixup

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5fbd7f6  alternative method of constructing plethysm branching rules

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7e13c8c  implemented maximal subgroups function

comment:55 Changed 5 years ago by
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21f0aea  minor docfixes

comment:56 Changed 5 years ago by
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faec227  moved branching code to branching_rules.py

comment:57 Changed 5 years ago by
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comment:59 Changed 5 years ago by
I'm looking at the code and its great. One worry that I have is that the global maximal_subgroups
can lead to confusion, as a user would probably expect
sage: maximal_subgroups(SymmetricGroup(4))
to work as well. Also, global functions can't be discovered through tab completion. Is there any reason for not making it a method like
sage: A4 = WeylCharacterRing('A4') sage: A4.maximal_subgroups()
Also, it is nice to be able to print a table of subgroups but the most useful thing in actual computations is to return objects that one can work with. So maybe make mode='get_rule'
the default?
comment:60 Changed 5 years ago by
There are two suggestions here. I think I agree with the first though there was a rationale for making maximal_subgroups global: this was that as it stands one may call it without creating the WeylCharacterRing?. So if you wanted to explore the subgroup lattice by descending through various subgroups this would be an advantage.
Still I agree with the comment.
For the second, if one is making maximal_subgroups a method, one might have two different methods and avoid the question of which is default. The question remains how to name them ...
comment:61 Changed 5 years ago by
I see. I don't have a particular suggestion for method names... maximal_subgroups()
vs. maximal_subgroups_table()
? A third option would be to write a custom container for the subgroups that behaves like a dictionary but prints like the table, though imho that would be overengineered for the problem at hand.
comment:62 followup: ↓ 64 Changed 5 years ago by
In the "Nonmaximal Levi subgroups and Projection from Reducible Types" section you say that 'branching_rule("E6","A5","levi") returns a notimplemented error'. But you did implement it in this patch ;) Its a good example, maybe just change it to say that it used to be not implemented (although it works now). Is there any Levi branching rule that still needs manual intervention? Maybe that subsection should be changed to just talk about projection to reducible types?
comment:63 followup: ↓ 65 Changed 5 years ago by
Is there an easy way to apply a BranchingRule
to a representation (i.e. a WeylCharacterRing.element_class
)? This seems like it would be one of the most common use cases:
sage: rule = branching_rule("A3", "C2", "symmetric") sage: rep = WeylCharacterRing("A3")(1,0,0,0) sage: rule.branch(rep) # does not exist sage: rep.branch(rule) # does not work sage: rule(rep) # somewhat unexpected result [0, 0]
The only thing I can come up with is the slightly awkward
sage: rep.branch(WeylCharacterRing(rule.Stype()), rule=rule) C2(1,0)
The BranchingRule.__call__
could check for the parent of the argument and simply return the branched representation.
comment:64 in reply to: ↑ 62 Changed 5 years ago by
Replying to vbraun:
Hopefully all branching rules to maximal subgroups are either implemented or constructable as plethysms, but branching to a nonmaximal Levi may or may not be implemented. For example, one of the G2 Levi's is implemented, the other not. (Neither Levi is a maximal subgroup.)
I'll have to correct the documentation per your comment.
In the "Nonmaximal Levi subgroups and Projection from Reducible Types" section you say that 'branching_rule("E6","A5","levi") returns a notimplemented error'. But you did implement it in this patch ;) Its a good example, maybe just change it to say that it used to be not implemented (although it works now). Is there any Levi branching rule that still needs manual intervention? Maybe that subsection should be changed to just talk about projection to reducible types?
comment:65 in reply to: ↑ 63 Changed 5 years ago by
Replying to vbraun:
sage: rep.branch(rule) # does not work
You do need to construct the WeylCharacterRing
you are branching to. The following works.
sage: br = branching_rule("A3", "C2", "symmetric") sage: [A3,C2] = [WeylCharacterRing(x,style="coroots") for x in ["A3","C2"]] sage: rep=A3(1,0,0,0); rep A3(1,0,0) sage: rep.branch(C2, rule=br) C2(1,0)
sage: rule.branch(rep) # does not exist
It does seem like an excellent idea for branch
to also be a method of the branching rule and I'll try to implement that.
comment:66 Changed 5 years ago by
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Branch pushed to git repo; I updated commit sha1. New commits:
4dac78b  implemented maximal_subgroup(s) as a WeylCharacterRing method etc.

comment:67 Changed 5 years ago by
 Reviewers set to Volker Braun
 Status changed from needs_review to needs_work
 Work issues set to Thematic tutorial needs further revision
New commits:
4dac78b  implemented maximal_subgroup(s) as a WeylCharacterRing method etc.

comment:68 Changed 5 years ago by
In response to the above discussion, WeylCharacterRing
instances now have two (very similarly named) methods, maximal_subgroups
and maximal_subgroup
. Also, the BranchingRule
class has a branch
method.
The thematic tutorials still need to be revised so I changed the status to needs_work.
comment:69 Changed 5 years ago by
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d6de77b  removed bogus commits

comment:70 Changed 5 years ago by
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f688ff3  reference: branching_rules: minor fixup

comment:71 Changed 5 years ago by
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7ca100a  revised thematic tutorial regarding maximal_subgroup(s) method

comment:72 Changed 5 years ago by
 Status changed from needs_work to needs_review
I have pushed revisions to the thematic tutorial. I'm changing the status back to needs_review
comment:73 Changed 5 years ago by
 Work issues Thematic tutorial needs further revision deleted
comment:74 Changed 5 years ago by
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Branch pushed to git repo; I updated commit sha1. New commits:
d3e1db5  note that the E6>A5 rule is now implemented

comment:76 Changed 5 years ago by
 Resolution set to fixed
 Status changed from needs_review to closed
The principal changes for this ticket are listed above. This comment contains a list of other less important changes.
In both the thematic tutorial and the docstring, hand written rules are discussed. It is omitted to say that the rule should map the positive Weyl chamber into the positive Weyl chamber. The doc should be revised on this point. (It is not *necessary* for the rule to map dominant weights into the positive Weyl chamber but it is desirable.)
In thematic tutorial and docstring, we should explain how to branch
B_r => D_r
andD_r => B_{r1
}. This should go in the section orthogonalandsymplecticsubgroupsoforthogonalandsymplecticgroups of the thematic tutorial. The r=4 case should be used as an example.Some doctests in weyl_characters are marked "long time" with obsolete timing information. For example the doctest [F4(fw).branch(B3,rule="levi") for fw in F4.fundamental_weights()] is given with the timing information 36s. This takes less than a second on my laptop. These misleading timings should be removed.
Branching (rule="levi") from E6 to A5 currently returns an error message, but this branching rule should be implemented.
Branching from composite type needs work: for example, branching from AxB to A or B should be implemented.