# Decomposing tensor powers of the fundamental representation of exceptional Lie algebras

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"?

• Probably there is no known answer. Anyway, keep in mind that there are six fundamental modules for $E_6$ (the rank being 6). Aug 1, 2019 at 1:41

• @JimHumphreys: ah, I missed that the asker was interested specifically in exceptional types. (Though I'd then maybe point out that "fundamental representation" could be misleading... probably for $E_6$ you'd want one of the two isomorphic minuscule representations?) Aug 1, 2019 at 1:41