Suppose $\chi$ is a irreducible character of a finite group G,show that $\chi(g^2)$ is a difference of characters.

Question

Suppose $\chi$ is a irreducible character of a finite group G,show that $\chi(g^2)$ is a difference of characters. It's clear that $\chi(g^2)$ is a class function, but I don't know how to prove that its inner product with other irreducible characters are all integers.

Answer 1

From Symmetric and exterior power of representation we have the formula $$\chi_V(g^2)=\chi_V(g)^2-2\chi_{\wedge^2V}(g)=\chi_{V\otimes V}(g)-2\chi_{\wedge^2V}(g),$$ i.e., it corresponds to the virtual representation $[V\otimes V]-2[\wedge^2V]$.