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Mathematics · Glossary

What is character?

Also known as: class function

Definition 5.5 University Mathematics — Year 3 · Chapter 5 — Representations of Finite Groups

The character of (V,ρ)(V, \rho) is χρ(g)=tr⁡ρ(g)\chi_\rho(g) = \operatorname{tr}\rho(g). It satisfies χρ(e)=dim⁡V\chi_\rho(e) = \dim V, χρ(hgh−1)=χρ(g)\chi_\rho(hgh^{-1}) = \chi_\rho(g) (traces are conjugation-invariant): characters are class functions — elements of the space CF(G)\mathcal{CF}(G) of functions constant on conjugacy classes, equipped with the Hermitian inner product

⟨φ,ψ⟩=1∣G∣∑g∈Gφ(g)‾ ψ(g).\langle \varphi, \psi\rangle = \frac1{\abs G}\sum_{g \in G} \overline{\varphi(g)}\,\psi(g).
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