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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 $g = e$

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

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