Symmetric and alternating-squares of linear representation

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Definition

Let $(\rho ,V)$ be a linear representation of a group $G$. Then we can define the linear representations $(S^2\rho ,S^{2}V)$ and $({\mathrm{\Lambda }}^2\rho ,{\mathrm{\Lambda }}^{2}V)$, the symmetric and alternating-squares of $(\rho ,V)$ respectively, by restricting the representation $(\rho \otimes \rho ,V\otimes V)$ of $G$ to the eigenspaces corresponding to the symmetric and alternating-squares respectively, that is,

$S^2\rho (g)(v\otimes w)=\rho (g)(v)\otimes \rho (g)w$ for $v\otimes w\in S^{2}V$, ${\mathrm{\Lambda }}^2\rho (g)(v\otimes w)=\rho (g)(v)\otimes \rho (g)w$ for $v\otimes w\in {\mathrm{\Lambda }}^{2}V$.

Characters of the symmetric and alternating-squares

For a representation $\rho$, write ${\chi }_{\rho }$ for its character.

Then

${\chi }_{S^2\rho }=\frac{1}{2}({\chi }_{\rho }(g{)}^2+{\chi }_{\rho }(g^2))$, and

${\chi }_{{\mathrm{\Lambda }}^2\rho }=\frac{1}{2}({\chi }_{\rho }(g{)}^2-{\chi }_{\rho }(g^2))$.