A classical result of Weyl on realization of finite-dimensional irreducible representation leads to a nice and quite explicit decomposition of the tensor power $\bigotimes^k V$ of the defining representation of $G$. The proof proceeds by finding the Schur dual, which in case of $GL(n)$ is just the group algebra of $S_k$ and in case of orthogonal and symplectic group is the group algebra of a braid group. One can write down explicit projectors on the appropriate subrepresentations. This work has been extended only recently to the tensor power of the seven-dimensional representation of $G_2$ by Huang and Zhu.

The question is whether one has similar result for tensor powers of any irreducible representation of $G$. I am mainly interested in the projectors onto irreducible summands of $\bigotimes^k \mathfrak{su} (n)$.