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 $\mathbb{C}$.
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1$\begingroup$ What's the reason for calling this Schur-Weyl duality? I don't see an action of a symmetric group. $\endgroup$– Qiaochu YuanFeb 20, 2014 at 17:35
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$\begingroup$ @QiaochuYuan The name is probably inappropriate but what I had in mind is that $M$ and $N$, and hence $M\otimes N$ can be realized inside the tensor algebra of $V\oplus V^*$ where $V$ is the natural $GL_n$ representation. Mixed tensor powers in this tensor algebra obviously carry an action of the symmetric group. So I figured that a possible answer will somehow be build around that. $\endgroup$– AlexFeb 20, 2014 at 18:57
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$\begingroup$ A basic issue is that the highest weight vectors aren't unique, whenever the tensor product multiplicities are $>1$. $\endgroup$– Allen KnutsonFeb 20, 2014 at 21:04
1 Answer
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.
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$\begingroup$ In old Dieudonne's exposition, there is also some treatment of a few other classic groups like $\text{O}_n{\mathbb C}$ $\dots$ Could you please add a couple of modern (textbook level) references in this direction. $\endgroup$ Feb 20, 2014 at 18:10