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characters

A character of a representation \(\phi: G\to V\) is defined as \begin{align*} \chi_\phi(g) := \operatorname{Tr}(\phi(g)) ,\end{align*} i.e. by taking the trace of the image of \(g\) in any basis.

Big result: An irreducible representation of a finite group \(G\) is completely determined by its character.