[with patch, needs review] exceptional and reducible type branching rules

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The patch trac_5794-revised.patch supercedes the other two patches. It applies cleanly on sage-3.4.2.rc0. Patch summary:

- new branching rules, including exceptional and reducible types
- all new rules have tests but slower F4 tests are # long time
- bugfix: reducible root system methods simple_roots etc. return families
- Cartan Types B, C and D allow low rank cases such as D2, B1 and C1
- WeylCharacterRings: new style="coroots" option gives output similar to published tables
- WeylCharacterRings: methods for rank, simple_roots etc.
- doc revision in weyl_characters.py

This patch contains #5831. If this patch is merged, that ticket may be closed.

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The last change indicates that I changed the owner from tbd to joyner. I don't remember doing that, and I don't see how I could have done it accidentally. Maybe someone else changed the owner, presumably wdj or mabshoff, but then trac shouldn't show that I did. I am puzzled by this.

Here are some comments about the G2=>A1 Levi branching rule. There is a branching rule G2=>A1xA1 (rule = "extended"). This is not implemented yet. Currently Weyl character rings are broke for reducible root systems. (I have a patch for that but it is not posted on trac yet.) I intend to implement branching to reducible root systems but first I want to do a few exceptional branching rules first before tackling the *many* cases of branching to reducible root systems.

So G2=>A1xA1 will come in a later patch but it is relevant here so I will discuss it.

In the branching rule G2=>A1xA1, the second A1 is almost but not quite the A1 in the G2=>A1 Levi branching rule. (The short root A1.) So it might seem that one should implement G2=>A1xA1 and then you would get the G2=>A1 (rule = "levi") branching rule. However this is not quite true. The A1 in the G2=>A1 branching rule is GL(2) and the A1 in A1xA1 is SL(2).

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I uploaded a second patch trac_5794-f4.patch which goes on top of the first. It implements branching rules F4=>B3 (levi), F4=>C3 (levi) and F4=>B4 (extended).

There is another extended rule F4=>C3xA1 (not implemented yet, but hopefully to be implemented later).

In contrast with G2, for F4, both Levi branching rules are redundant since the Levi subgroups are not maximal. They factor through branching rules F4=>B4=>B3 and F4=>C3xA1=>C3. However I implemented them for convenience. You can check directly that F4(x).branch(B3,rule="levi") and F4(x).branch(B4,rule="extended").branch(B3,rule="levi") return the same thing for x in F4.fundamental_weights().

I compared the output for these rules against those that I could find in a book, Patera and Sankoff, Branching rules for representations of simple Lie algebras.

[mabshoff]

Hi Dan,

the change in ownership happend because you changed the component to "group_thoery". For every ticket you work on and post patches you should accept it (see the bottom left), that way ownership stays with you.

Cheers,

Michael