TITLE: Exceptional Schur-Weyl Duality
QUESTION [7 upvotes]: Are there any known analogs of Schur-Weyl duality for the exceptional groups?
In particular, I am looking to decompose the tensor powers of the action of $E_6$ on its 27-dimensional module.  Any advice or references to this end would be greatly appreciated.

REPLY [3 votes]: There is an article by Jing-Song Huang and Chen-Bo Zhu called Weyl's construction and tensor power decomposition for G2 that builds on an invariant theory for $G_2$ (see the linked review for details). There's quite recent article Algèbres de Jordan et théorie des invariants which deals with these invariants also for other exceptional cases and I would guess there'll be some others.