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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"?

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

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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).

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  • $\begingroup$ These are useful references for certain classical types, but the question is about the exceptional types of simple Lie algebras. Here there are limits to the knwn computational methods, but no analogue yet of Schur-Weyl theory. $\endgroup$
    – Jim Humphreys
    Commented Aug 1, 2019 at 1:31
  • $\begingroup$ @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?) $\endgroup$
    – Sam Hopkins
    Commented Aug 1, 2019 at 1:41
  • $\begingroup$ @ Jim, Sam Thanks for the comments. Is there a reason why this problem has proved so difficult? $\endgroup$
    – Nadia SUSY
    Commented Aug 2, 2019 at 11:36
  • $\begingroup$ I think partially the reason is because the answer itself is complicated, at least compared to the simplicity of Schur-Weyl duality. $\endgroup$
    – Sam Hopkins
    Commented Aug 2, 2019 at 15:10

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