For the $A$-series, tensor powers of the fundamental representation of $\frak{sl}\_n$ decompose into irreducibles according to a certain Young diagram/ partition formula. This inspires, for example, the theory of Schur functors.


What happens for the exceptional Lie algebras? For example, taking $V$, the fundamental representation of $E\_6$, do we have a formula for the decomposition of its tensor powers? Is there a theory of "exceptional Schur functors"?



Decomposing tensor powers of the fundamental representation of exceptional Lie algebras
If you're e.g. looking for an analog of Schur-Weyl duality in other types, the combinatorics can get very tricky very quickly. For the symplectic group I believe the answer was first worked out by Sundaram in her PhD thesis (<https://dspace.mit.edu/handle/1721.1/15060>) and meanwhile for the odd orthogonal group the answer was only obtained very recently by Jagenteufel (<https://arxiv.org/abs/1902.03843>).


