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

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?

`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). $\endgroup$ – Gro-Tsen Mar 7 '19 at 23:12