One can decompose $\bigotimes^2 V = \bigvee^2 V \oplus \bigwedge^2 V$, getting a corresponding decomposition for representations, say when $V$ is a module for some finite group $G$. One then has the relation between the characters $\chi$, $\vee^2 \chi$ and $\wedge^2 \chi$ afforded by the $G$-modules $V$, $\bigvee^2 V$ and $\bigwedge^2 V$, given by $(\wedge^2 \chi)(g) = \frac{1}{2} (\chi(g)^2-\chi(g^2))$ and $(\vee^2 \chi)(g) = \frac{1}{2} (\chi(g)^2 + \chi(g^2))$.
More generally, one can decompose $\bigotimes^r V$ by Schur-Weyl duality in terms of the irreducible representations of the symmetric group $S_r$. Can we use this to give similar formulas for the representations coming up in this decomposition?