Let \(G\) be a group, let \(k\) be a field, and let \(M\) be a finite-dimensional representation of \(G\) over \(k\), that is, a finite-dimensional \(k\)-vector space with a linear action \(G\actson M\). For \(g\in G\), let \[ T_g\colon M \to M, \qquad m \mapsto g\actson m. \] The character of \(M\) is the function \(\chi_M\colon G\to k\), \(\chi_M(g) \coloneqq \tr T_g\).
Character of a representation
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