Opened 13 years ago

Last modified 6 years ago

#5794 closed enhancement

[with patch, needs review] G2 and F4 branching rules — at Version 5

Reported by: bump Owned by: joyner
Priority: major Milestone: sage-4.3
Component: group theory Keywords: lie groups
Cc: sage-combinat Merged in:
Authors: Reviewers:
Report Upstream: N/A Work issues:
Branch: Commit:
Dependencies: Stopgaps:

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Description (last modified by bump)

This first patch implements branching rules for the following inclusions of Lie groups:

G2 => A1 (along short root) 
G2 => A2
B3 => G2
D4 => G2

The second patch implements branching rules:

F4 => B3
F4 => C3
F4 => B4

They go on top of the following patches:

trac_5721-a.patch
trac_5721-b.patch
trac_5751.patch

Change History (5)

comment:1 Changed 13 years ago by bump

  • Cc sage-combinat added
  • Component changed from algebra to group_theory
  • Keywords lie groups added
  • Owner changed from tbd to joyner

comment:2 Changed 13 years ago by bump

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).

comment:3 Changed 13 years ago by bump

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.

comment:4 Changed 13 years ago by bump

  • Summary changed from [with patch, needs review] G2 branching rules to [with patch, needs review] G2 and F4 branching rules

comment:5 Changed 13 years ago by bump

  • Description modified (diff)
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