I want to find explicit formulae for projectors onto irreducible summands of $\bigotimes^k \mathfrak{g}$, where $\mathfrak{g}$ is a simple Lie algebra. Or more generally, write down the projectors of $\bigotimes^k \mathbb{V}_\lambda$ for irreducible representation of $\mathfrak{g}$ of weight $\lambda$.

What I have found out so far:

A classical result of Weyl on realization of finite-dimensional irreducible representation leads to a nice and quite explicit decomposition of the tensor power $\bigotimes^k \mathbb{C}^n$ of the defining representation of $GL(n,\mathbb{C})$. A key point to the proof is the identification of the centralizer of the action inside $End(\bigotimes^k \mathbb{C}^n)$ (i.e. a commutant subalgebra). This subalgebra is actually a group algebra of the symmetric group $S_k$ which acts by permuting the vectors in the tensor product. Understanding of representation of this algebra helps to understand the multiplicities of of irreducible summands of $\bigotimes^k \mathbb{C}^n$. (A special case of Schur functors I suppose.) All of this can be found in the book by Goodmann and Wallach. There, one can also find a similar treatment for tensor powers of the defining representations of orthogonal and symplectic groups. The commutant subalgebra is then the Brauer algebra and again, the projectors are written down more or less explicitly. By the way, only recently the tensor power of the seven-dimensional representation of $G_2$ was decomposed in a similar manner by Huang and Zhu.

What else is known in this area? Can this approach be generalized to tensor powers of other representations?

Specifically, I want to understand a certain ideal in the universal enveloping algebra of $\mathfrak{su}_n$ and I am trying to find its generators by determining which irreducible summands belongs to the ideal and which not. Accordingly, I am mainly interested in the decomposition of $\bigotimes^k \mathfrak{su}_n$.

  • $\begingroup$ Your last symbol looks confusing. If you are concerned about finite dimensional irreducible representations of compact Lie groups, these are essentially the "same" as the corresponding representations of complex groups. I'm not sure which $G$` you are interested in. $\endgroup$ Jan 13 '11 at 15:37
  • $\begingroup$ Also "Schur projectors" as a poor choice of name, since I think most people would use that to mean the projections onto the Schur functors en.wikipedia.org/wiki/Schur_functor which I don't think is what you mean, though you are actually a bit vague about what you mean. $\endgroup$
    – Ben Webster
    Jan 13 '11 at 21:23
  • $\begingroup$ I've rewritten my question. Since $\mathfrak{g}$ is more or less $V \otimes V^*$ with an invariant pairing I guess that this cannot be much different from the orthogonal case of the classical Schur-Weyl theory. $\endgroup$ Jan 14 '11 at 19:08
  • $\begingroup$ There are some nice tables of decompositions of tensor products of irreducible representations in: E.B. Vinberg and A.L. Onishchik, Seminar on Lie groups and algebraic groups, Springer, 1990 (?), but they probably contain only very partial data comparing with what you are interested in. $\endgroup$ May 9 '11 at 6:44

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