Exercise from F+H, Exercise 1.3:
Let $\rho : G \rightarrow GL(V)$ be any representation of the finite group $G$ on a $n$-dimensional vector space $V$ and suppose that for any $g \in G$ the determinant of $\rho(g)$ is 1. Show that $\bigwedge^k V$ and $\bigwedge^{n-k} V^*$ are isomorphic as representations of G
For the life of me I can't figure out. I know that $\bigwedge^k V$ and $\bigwedge^{n-k} V^*$ are isomorphic as spaces, but why are they isomorphic as representations, I have no idea. I suspect it has something to do with the determinant being 1, but...