Character (mathematics)

In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings.^[1] Other uses of the word "character" are almost always qualified.

Multiplicative character

Main page: Multiplicative character

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 (thus the 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 [math]\displaystyle{ \chi_1,\chi_2, \ldots , \chi_n }[/math] are different characters on a group G then from [math]\displaystyle{ a_1\chi_1+a_2\chi_2 + \dots + a_n \chi_n = 0 }[/math] it follows that [math]\displaystyle{ a_1=a_2=\cdots=a_n=0 }[/math].

Character of a representation

Main page: Character theory

The character [math]\displaystyle{ \chi : G \to F }[/math] of a representation [math]\displaystyle{ \phi \colon G\to\mathrm{GL}(V) }[/math] of a group G on a finite-dimensional vector space V over a field F is the trace of the representation [math]\displaystyle{ \phi }[/math] (Serre 1977), i.e.

[math]\displaystyle{ \chi_\phi(g) = \operatorname{Tr}(\phi(g)) }[/math] for [math]\displaystyle{ g \in G }[/math]

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.

Alternative definition

If restricted to finite abelian group with [math]\displaystyle{ 1 \times 1 }[/math] representation in [math]\displaystyle{ \mathbb{C} }[/math] (i.e. [math]\displaystyle{ \mathrm{GL}(V) = \mathrm{GL}(1, \mathbb{C}) }[/math]), the following alternative definition would be equivalent to the above (For abelian groups, every matrix representation decomposes into a direct sum of [math]\displaystyle{ 1 \times 1 }[/math] representations. For non-abelian groups, the original definition would be more general than this one):

A character [math]\displaystyle{ \chi }[/math] of group [math]\displaystyle{ (G, \cdot) }[/math] is a group homomorphism [math]\displaystyle{ \chi: G \rightarrow \mathbb{C}^* }[/math] i.e. [math]\displaystyle{ \chi (x \cdot y)=\chi (x) \chi (y) }[/math] for all [math]\displaystyle{ x, y \in G. }[/math]

If [math]\displaystyle{ G }[/math] is a finite abelian group, the characters play the role of harmonics. For infinite abelian groups, the above would be replaced by [math]\displaystyle{ \chi: G \to \mathbb{T} }[/math] where [math]\displaystyle{ \mathbb{T} }[/math] is the circle group.

References

↑ "character in nLab". https://ncatlab.org/nlab/show/character.

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
Serre, Jean-Pierre (1977), Linear Representations of Finite Groups, Graduate Texts in Mathematics, 42, Translated from the second French edition by Leonard L. Scott, New York-Heidelberg: Springer-Verlag, doi:10.1007/978-1-4684-9458-7, ISBN 0-387-90190-6, https://archive.org/details/linearrepresenta1977serr