Multiplicative character

In

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

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

• Consider the (ax + b)-group

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

Functions f_u : G → C such that ${\displaystyle f_{u}\left({\begin{pmatrix}a&b\\0&1\end{pmatrix}}\right)=a^{u},}$ 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.

• ISBN 978-0-486-62342-9
  Lectures Delivered at the University of Notre Dame