# orthogonality relation for characters

Let $$\rho : G \to GL(V)$$ be a representation on G. Then, its character is defined as $$\chi_V(g) := Tr(\rho(g))$$, where $$Tr$$ denotes the trace function.

For an exercise I am trying to solve, I would like to have the following relation without using directly Schur relations, because I did not have these in lecture.

$$\frac{1}{|G|} \sum_{\rho \in Irr(V)} \chi_{\rho}(e) \chi_{\rho}(h) = 0$$ for $$g \neq e$$ and $$\frac{1}{|G|} \sum_{\rho \in Irr(V)} \chi_{\rho}(e) \chi_{\rho}(h) = 1$$ for $$g = e$$

• this is in D&F in the chapter on rep theory if you have that to look it up in – Alexander Gruber Jan 8 at 22:12