Characters

Abstract

The character of a representation ρ of a group G on a finite-dimensional vector space E, over a field $\mathbb{F}$, is the function χ_ρ on G given by

$${\chi }_{\rho } : G \rightarrow\mathbb{F} : g\mapsto \mathrm{Tr}{\bigl (\rho (g)\bigr )}.$$

(7.1)

This logically sensible definition turns history on its head, because Frobenius [29] first constructed characters directly from groups, and certain associated determinants, and not from their representations.