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Character theory studies a group representation through the trace of each representing operator. For $\rho :G\to \mathrm{GL}(V)$, the character is $\chi (g)=\mathrm{tr}\rho (g)$, a class function because trace is invariant under conjugation.

For finite groups over $\mathbb{C}$, character orthogonality recovers irreducible multiplicities: $⟨{\chi }_V,{\chi }_W⟩$ measures the number of shared irreducible constituents with multiplicity. A permutation representation has character equal to the number of fixed points, making abstract traces directly countable. Characters therefore compress large matrices into one value per conjugacy class while preserving the isomorphism class of a complex representation.

A finite group of Lie type is obtained from rational points of a connected reductive algebraic group with a suitable Frobenius endomorphism of an algebraic group, often followed by central quotient or passage to a derived subgroup. Its complex representations combine finite-group character theory with the geometry of Borel subgroups, maximal algebraic tori, and the Weyl group.