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][1]. 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][2] 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][3].

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 $\bigotimes^k \mathfrak{su}_n$. 

  [1]: http://books.google.com/books?id=tbSX5VPE4PIC&lpg=PA528&dq=goodmann%2520wallach&pg=PA528#v=onepage&q&f=false
  [2]: http://en.wikipedia.org/wiki/Brauer_algebra
  [3]: http://www.jstor.org/stable/119028