 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.
Contents
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 quasicharacters. Dirichlet characters can be seen as a special case of this definition.
Character of a representation
Main article: Character theoryThe character of a representation φ of a group G on a finitedimensional vector space V over a field F is the trace of the representation φ. In general, the trace is not a group homomorphism, nor does the set of traces form a group. The characters of onedimensional representations are identical to onedimensional 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.
See also
 Dirichlet character
 HarishChandra character
 Hecke character
 Infinitesimal character
 Alternating character
References
 Artin, Emil (1966), Galois Theory, Notre Dame Mathematical Lectures, number 2, Arthur Norton Milgram (Reprinted Dover Publications, 1997), ISBN 9780486623429
 Serre, JeanPierre (1977), Linear Representations of Finite Groups, SpringerVerlag, ISBN 0387901906.
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