Character (mathematics) - Character of A Representation

The character of a representation φ of a group G on a finite-dimensional vector space V over a field F is the trace of the representation φ (Serre 1977). In general, the trace is not a group homomorphism, nor does the set of traces form a group. The characters of one-dimensional representations are identical to one-dimensional representations, so the above notion of multiplicative character can be seen as a special case of higher dimensional characters. The study of representations using characters is called "character theory" and one dimensional characters are also called "linear characters" within this context.