I want to show that $S_n$ has only two 1 dimensional represnetations. mainly the trivial and sign represnetations.
Where I assumed that our Field we're working on is with characteristic $\neq 2$.
What I wrote so far, that if I assume that there exists another nonequivalent represnetation to the above representations then (denote it by $\tau$, and $\chi_{\tau} = trace\ \tau $):
$\sum_{\sigma \in A_n} \chi_{\tau}(\sigma) = \sum_{\sigma \in S_n -A_n} \chi_{\tau} (\sigma) =\sum_{\sigma \in S_n} \chi_{\tau} (\sigma) = 0 $
Somehow, I want to derive some sort of contradiction, I thought of deriving that $\dim \tau =0$, but don't see exactly how?
I mean I know from above that:
$ 1=\dim \tau = -\sum_{\sigma \in A_n \ \sigma \neq id} \chi_{\tau} (\sigma)$
But other than that, I don't see how to derive a contradiction?
Any help is appreciated