The dimension of $S_{21}(V)$ is $(n+1)n(n-1)/3$; it is only $8$ if $n=3$. The other half of Schur Weyl duality says that $S_{21}(V)$ is a $GL_n$ (not $S_n$) S_3$) irrep; namely, the one which is indexed by the partition $(2,1)$. Similarly, $\mathrm{Sym}^3(V)$ and $\bigwedge^3 V$ have dimensions $(n+2)(n+1)n/6$ and $n(n-1)(n-2)/6$ and are $GL_n$, not $S_3$, irreps.
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The dimension of $S_{21}(V)$ is $(n+1)n(n-1)/3$; it is only $8$ if $n=3$. The other half of Schur Weyl duality says that $S_{21}(V)$ is a $GL_n$ (not $S_n$) irrep; namely, the one which is indexed by the partition $(2,1)$. Similarly, $\mathrm{Sym}^3(V)$ and $\bigwedge^3 V$ have dimensions $(n+2)(n+1)n/6$ and $n(n-1)(n-2)/6$ and are $GL_n$, not $S_3$, irreps. |
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