I am having trouble understanding part of a lecture in class.
Let $G$ be a group, $\phi$ a representation of $G$ into $GL_n(C)$, and let $E^{ij}$ denote the matrix with 1 in the $i$th row and $j$th column and zero everywhere else.
Define the matrix $M = \sum_{g \in G} \phi(g) E^{ij} \phi(g)^{-1}$. We showed that $M$ commutes with $\phi(g)$ for every $g$. So by Schur's lemma, we know that $M = \alpha I_n$ for some nonzero complex number $\alpha$.
However, if you compute the trace of $M$, it comes out to be $|G| \delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta function. Since $M = \alpha I_n$, the trace should simply be a fixed number $\alpha n$.
Where is the mistake?