# Linear character

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with Character of a linear representation

This term is related to: linear representation theory
View other terms related to linear representation theory | View facts related to linear representation theory

## Definition

A linear character of a group over a field is a homomorphism from the group to the multiplicative group of the field. Equivalently, it can be thought of as a representation of the group over the field, whose degree (dimension) is one.

People working with Abelian groups typically use the term character to refer to a linear character over the complex numbers that always take values on the unit circle (in other words, a unitary linear character). For them, a character is thus simply a homomorphism to the circle group.

## Study of this notion

### Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20C