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?



Branching to Levi subgroups in SAGE and the circle action
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](http://doc.sagemath.org/html/en/thematic_tutorials/lie/branching_rules.html#levi-subgroups) 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.)


