# Branching to Levi subgroups in SAGE and the circle action

In the SAGE computer package, there useful exist tools for branching representations of a simple Lie group to a Levi subgroup:

http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/branching_rules.html#sage.combinat.root_system.branching_rules.BranchingRule.Stype

Explicitly, one is branching to subgroup corresponding to a Dynkin sub-diagram, obtained by removing a single node.

For example, we can branch from $$SL(n)$$ to the subgroup $$SL(n-1)$$. However, $$SL(n-1)$$ can be considered as "living" in the larger subgroup $$SL(n-1) \times U1$$. This is true for every subgroup coming from a deleted node, i.e. one can always take the product of the subgroup with U1, to obtain a larger subgroup.

How does one branch to this subgroup in SAGE. For example, it is done in the LieArt program for mathematica: see A3 of the following

https://arxiv.org/pdf/1206.6379.pdf

Is this also possible in SAGE?

• The case of $A_n$ is a bit special: if I recall correctly, SAGE actually deals with $\mathit{GL}_{n+1}$, not $\mathit{SL}_{n+1}$ when it says $A_n$ (perhaps only if you don't set style="coroots"). So in this particular case there's probably a cheap way to get the answer (but there may be a better one also). Otherwise, I think I recall LiE can do the computation (and it can be called from SAGE but maybe that's considered cheating). – Gro-Tsen Mar 7 '19 at 23:12

Branching to Levi subalgebras should really take into account the central part of the Levi subalgebra but it is not the case. The problem is that the WeylCharacterRing is defined only for semisimple Lie algebras and not for reductive ones. The mathematical side of the problem is easy -- decompose orthogonally your Cartan algebra into "central part" and "semisimple part". For semisimple part you have the branching there already. For central part, nothing really happens. In type A that is pretty much what the method coerce_to_sl is doing -- i.e. it is a projector onto the semisimple part.

But then again, as was noted by Gro-Tsen in his comment, for type A, if you work in the Bourbaki basis and not in basis of fundamental weights, then you are effectively working with $$GL$$s instead of $$SL$$s. See Sage tutorial on branching for example.

Extension of WeylCharacterRing so that it can handle also reductive algebras is on my long term TODO list so this kind of branching may end up being integral part of Sage in the future. (No promises though.)