I have this algebra task which I have encountered problems with proving a specific identity for,
Consider the permutation representation of the symmetric group $_$, which gives a group homomorphism $\rho : _ \to (\mathbb{C}^n)$ and is defined by $$ \rho()_=_{()}, $$where $_1,…,_$ are the standard basis vectors of $\mathbb{C}^n$.
I'm asked to show that $tr(\rho())$ is equal to the number of $ \in \{1,…,\}$ such that $()=$.
I would appreciate any helps in this regards as I have no idea where to even start.