Let $ \chi $ be the character of an irrep of a finite group $ G $. Let $ \pi $ be a representation of $ G $. Then the projection onto the $ \chi $ isotypic subrepresentation is given by $$ \Pi_\chi= \frac{degree(\chi)}{|G|} \sum_{g \in G} \overline{\chi(g)} \pi(g) $$ Let $ \chi_1 \neq \chi_2 $ be characters of distinct irreps of $ G $. How do you show that $$ \Pi_{\chi_1} \Pi_{\chi_2}=0 $$
I know the above equation must be be true but lately I've been looking at these irrep characters of $ SL(2,3) $ $$ \chi_1=(2 ,-2, 0, -\omega, -\omega^2, \omega^2, \omega) \\ \chi_2=(2 ,-2, 0, -\omega^2, -\omega, \omega, \omega^2) $$ For more details on these characters see
https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_special_linear_group:SL(2,3)
And I have not been getting $$ \Pi_{\chi_1} \Pi_{\chi_2}=0 $$ What is going wrong?