# Exceptional Schur-Weyl Duality

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.

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Schur-Weyl duality can be obtained via the more general notion of Howe duality of reductive pairs (in this case, GLn and GLm). It might be useful to look at R. Howe's 'Schur Lectures' where this framework is discussed; these notes should be available at a good library and there is also a copy floating around online somewhere. –  George Melvin Feb 27 at 22:27
... Here's the link math.ethz.ch/~khorosh/teaching/sym_functions/list.html –  George Melvin Feb 27 at 22:29
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## 1 Answer

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.

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your second link goes to a paper of Deligne and Gross instead of that by Bruno Blind. –  Abdelmalek Abdesselam Feb 27 at 23:33
Already fixed. Thanks. –  Vít Tuček Feb 27 at 23:34
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