# Math Help - Character Theory II

1. ## Character Theory II

Think back to what irreducible characters are--they are the characters of irreps on $G$. So, let $\chi\in\text{irr}(G)$ and let $\rho:G\to\text{GL}\left(V\right)$ be the irrep it came from. Then, as you can easily verify $\chi^a$ as you denoted is the character of $\rho\circ a:G\to\text{GL}(V)$ (which is a representation since $a\in\text{End}(G)$). Now, if $\rho\circ a$ left any non-trivial proper subspace of $V$ $\rho\circ a$-invariant wouldn't it be necessary (using the fact that $a$ is bijective) that same subspace would be $\rho$-invariant? So...