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.