= Character theory
{wiki}
<Character theory> studies a <group representation> through the <trace> of each representing operator. For $\rho:G\to\operatorname{GL}(V)$, the <character> is $\chi(g)=\operatorname{tr}\rho(g)$, a <class function> because trace is invariant under conjugation.
For <finite groups> over $\mathbb C$, <character orthogonality> recovers irreducible multiplicities: $\langle\chi_V,\chi_W\rangle$ 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>.
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