Character (mathematics)


In mathematics, a character is a special kind of function from a group to a field. There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualified.

Multiplicative character

A multiplicative 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 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 ; 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 are different characters on a group G then from it follows that.

Character of a representation

The character of a representation of a group G on a finite-dimensional vector space V over a field F is the trace of the representation , i.e.
for
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 representation in , the following alternative definition would be equivalent to the above :
A character of Group is a mapping such that for all
If is a finite Abelian group, the characters play the role of harmonics. For infinite Abelian Group, the above would be replaced by where is the Circle group.