Character (mathematics)
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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.
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[edit] 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 an abelian group, then the set Ch(G) of these morphisms forms a group under pointwise multiplication.
This group is referred to as the character group of G. Sometimes only unitary characters are considered (so that 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.
[edit] Character of a representation
The character of a representation φ of a group G on a vector space V over a field F is the trace of the representation φ (Serre 1977). 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.
[edit] See also
- Dirichlet character
- Harish-Chandra character
- Hecke character
- Infinitesimal character
- Alternating character
[edit] References
- Artin, Emil (1966), Galois Theory, Notre Dame Mathematical Lectures, number 2, Arthur Norton Milgram (Reprinted Dover Publications, 1997), ISBN 978-0486623429
- Serre, Jean-Pierre (1977), Linear Representations of Finite Groups, Springer-Verlag, ISBN 0-387-90190-6.
