Let V be a complex représentation of finite dimension of the symmetric group $S_n$ and $χ$ the character of $V$.
prove $V=V\otimes_{\Bbb C}\varepsilon$ iff $χ(\sigma)=0$ for every odd permutation, where $\varepsilon$ is the signature.
The prove is straightforward $χ(\sigma) = χ(\sigma) \varepsilon(\sigma) = - χ(\sigma) \Leftrightarrow χ(\sigma)=0$.
But I am not sure if $χ(\sigma) = χ(\sigma) \varepsilon(\sigma)\Leftrightarrow V=V\otimes_{\Bbb C}\varepsilon $ ?
Thank you for your help.