TITLE: Schur-Weyl duality
QUESTION [7 upvotes]: I have a question about the interpretation of multiplicities and dimensions using Schur-Weyl duality.
$V$ is an n-dimensional complex vector space. Then $V$ $\otimes$ $V$ $\otimes$ $V$ decomposes as:
$V$ $\otimes$ $V$ $\otimes$ $V$  = $Sym^3$ $V$ $\oplus$ $\wedge^3$ $V$ $\oplus$  $S_{(2,1)}V$  $\oplus$  $S_{(2,1)}V$
where $S_{(2,1)}V$ is a Schur module for the partition (2,1). (Fulton-Harris, Chapter 6).
Then Schur-Weyl duality says that the multiplicity of $S_{(2,1)}V$ in this decomposition (it is 2) should be the dimension of the irrep of $S_3$ labelled by the partition (2,1) - which is correct.
My question is about the other half of this duality: the dimension of $S_{(2,1)}V$ in this decomposition (it is 8) should correspond to some sort of multiplicity for the irrep of $S_3$ labelled by the partition (1,2) - but I am unable to see exactly what...
Any help is most welcome.

REPLY [3 votes]: Let $V$ be the vector representation of $GL_n(\mathbb{C})$, and let $d\leq n$. 
You want to see the multiplicities of a given irreducible $S_d$ module in $V^{\otimes d}$ in terms of the dimension of an associated irreducible $GL_n(\mathbb{C})$-module. To do this, observe that $S_d$ acts on $V^{\otimes d}$ by permuting the tensor factors and this action commutes with the diagonal action of $GL_n(\mathbb{C})$ on $V^{\otimes d}$. Therefore, for a partition $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n\geq 0)$ (equivalently, a polynomial weight for $GL_n(\mathbb{C})$) we can consider the space $W_\lambda$ of highest weight vectors in $V^{\otimes d}$ of highest weight $\lambda$. Since the action of $S_d$ commutes with the action of $GL_n(\mathbb{C})$ (and, in particular, the action of the standard maximal torus, and standard Borel), $W_\lambda$ is an $S_d$-module. By character considerations, one can show that this space is the irreducible Specht module, $W_\lambda\cong S^{\lambda}$. 
More generally, if $X\in GL_n(\mathbb{C})$, then $X\cdot W_\lambda$ is another $S_d$-module since, again, the action of $S_d$ commutes with the action of $GL_n(\mathbb{C})$. Moreover, this module is obviously isomorphic to $S^\lambda$. This means that there are $\dim L(\lambda)$ copies of $S^\lambda$ in $V^{\otimes d}$, where $L(\lambda)$ is the irreducible $GL_n(\mathbb{C})$-module of highest weight $\lambda$. 
This proves half of the statement that, as a $(GL_n(\mathbb{C}),S_d)$-bimodule,
$V^{\otimes d}\cong \bigoplus_{\lambda\vdash d}L(\lambda)\otimes S^\lambda.$ From your question, I assume you understand the other half.