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. Other uses of the word "character" are almost always qualified.
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 , 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 formula_1 are different characters on a group "G" then from formula_2 it follows that formula_3.
Character of a representation.
The character formula_4 of a representation formula_5 of a group "G" on a finite-dimensional vector space "V" over a field "F" is the trace of the representation formula_6 , i.e. 
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 formula_9 representation in formula_10 (i.e. formula_11), the following alternative definition would be equivalent to the above (For abelian groups, every matrix representation decomposes into a direct sum of formula_9 representations. For non-abelian groups, the original definition would be more general than this one):
A character formula_13 of group formula_14 is a mapping formula_15 such that formula_16 for all formula_17
If formula_18 is a finite abelian group, the characters play the role of harmonics. For infinite abelian groups, the above would be replaced by formula_19 where formula_20 is the circle group.