Split $V \otimes V$ into simple $S_2$-representation.

Question

Let $V$ be a $\mathbb{C}$-vector space. Let the symmetric group $S_2$ act on $V \otimes V$ by permuting the factors. Split $V \otimes V$ into simple $S_2$-representation.

A simple (sub-)representation would be a vector space (subspace) that is invariant under the action of $S_2$. So would the simple representations be $W_{ij} = span\{v_i \otimes v_j, v_j \otimes v_i\}$, where $v_i$ are the basis vectors for $V$?

Answer 1

Note that if $i \neq j$ then the $\mathbb{C}$-span of $v_i \otimes v_j + v_j \otimes v_i$ is an $S_2$-invariant subspace. Similarly for $v_j \otimes v_i - v_i \otimes v_j$. You need to take these 1-dimensional subspaces into account.