I know that for any vector space $V$, $$\textstyle V \otimes V \cong \operatorname{Sym}^2 V \oplus \bigwedge^2 V.$$ But I was going trough some lecture notes on representation theory and encountered the following exercise.
Exercise: Show that the map $\theta$ defined on $V\otimes V$ by $e_ i\otimes e_j\mapsto e_j\otimes e_i$ gives rise to the $G$-decomposition $V \otimes V \cong \operatorname{Sym}^2 V \oplus \bigwedge^2 V$. Use this to calculate the characters of $\operatorname{Sym}^2 V$ and $\bigwedge ^2 V$.
I know how to prove the decomposition by another mean, namely writing $v\otimes w$ as $$\frac{1}{2}(v\otimes w+w\otimes v)+\frac{1}{2}(v\otimes w-w\otimes v)$$ and calculate the characters of the spaces by considering the eigenvalues of each space. But I do not understand how the author intends us to use the map $\theta$. What does he mean by it gives rise to the decomposition? How should I use it to calculate the characters?
