Multiplicative character

In mathematics, a multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field (Artin 1966), usually the field of complex numbers. If G is any group, then the set Ch(G) of these morphisms forms an abelian group under pointwise multiplication.

This group is referred to as the character group of G. Sometimes only unitary characters are considered (characters whose image is in the unit circle); other such homomorphisms are then called quasi-characters. Dirichlet characters can be seen as a special case of this definition.

Multiplicative characters are linearly independent, i.e. if χ 1 , χ 2 , … , χ n {\displaystyle \chi _{1},\chi _{2},\ldots ,\chi _{n}} are different characters on a group G then from a 1 χ 1 + a 2 χ 2 + ⋯ + a n χ n = 0 {\displaystyle a_{1}\chi _{1}+a_{2}\chi _{2}+\cdots +a_{n}\chi _{n}=0} it follows that a 1 = a 2 = ⋯ = a n = 0. {\displaystyle a_{1}=a_{2}=\cdots =a_{n}=0.}

• Consider the (ax + b)-group

G := { ( a b 0 1 )   |   a > 0 ,   b ∈ R } . {\displaystyle G:=\left\{\left.{\begin{pmatrix}a&b\\0&1\end{pmatrix}}\ \right|\ a>0,\ b\in \mathbf {R} \right\}.}

Functions f_u : G → C such that f u ( ( a b 0 1 ) ) = a u , {\displaystyle f_{u}\left({\begin{pmatrix}a&b\\0&1\end{pmatrix}}\right)=a^{u},} where u ranges over complex numbers C are multiplicative characters.

• Consider the multiplicative group of positive real numbers (R⁺,·). Then functions f_u : (R⁺,·) → C such that f_u(a) = a^u, where a is an element of (R⁺, ·) and u ranges over complex numbers C, are multiplicative characters.

• Artin, Emil (1966), Galois Theory, Notre Dame Mathematical Lectures, number 2, Arthur Norton Milgram (Reprinted Dover Publications, 1997), ISBN 978-0-486-62342-9 Lectures Delivered at the University of Notre Dame