Schur-Weyl duality for arbitrary tensor products of simple finite-dimensional $GL_n$-modules Let $M$ and $N$ be two simple finite dimensional $GL_n$ modules. Is there a way of expressing the heighest weight vectors of the simple submodules of $M\otimes N$ in terms of the heighest weight vector of $M$ and $N$ and (possibly) Young projection operators? I couldn't find anything in Fulton's book, but I guess this has to be known (or not?).To avoid confusion, I work over $\mathbb{C}$. 
 A: The most generic answer that occurs to me is to use the Littlewood-Richardson Rule for writing a product of Schur polynomials as a weighted sum of Schur polynomials. The idea is to realize each irreducible representation of the general linear group as the result of applying the Schur functor $\mathcal{F}$ to a Specht module. More explicitly, $M=V({\lambda})$ and $N=V({\mu})$, where $\lambda$ and $\mu$ are the highest weights, namely partitions of length $\leq n$. We then have $$\mathcal{F}(S^{\lambda})\cong V(\lambda)$$ and $$\mathcal{F}(S^{\mu})\cong V(\mu).$$ Then, decompose the "product" $[S^{\lambda}][S^{\mu}]$ (really, an induced representation) of these Specht modules in the full representation ring for all symmetric groups into Specht modules. This is best accomplished by using the Littlewood-Richardson Rule, since we have an isomorphism between the full representation ring and the ring of symmetric functions. The isomorphism sends $[S^{\nu}]$ to the Schur polynomial $s_{\nu}$. So, the exercise is really to decompose the product $s_{\lambda}s_{\mu}$ into a weighted sum of Schur polynomials by using the Littlewood-Richardson Rule. The partitions of the Schur polynomials occurring in the sum are then the highest weights.
