Character (mathematics)
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There are several meanings of the word character in mathematics, although all are related to the idea of using fields (most of the time the complex numbers), to study a more abstract algebraic structure.
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[edit] Number-theoretic characters
If G is group, a character is a group homomorphism into the multiplicative group of a field (as defined in Emil Artin's book on Galois Theory), usually the field of complex numbers. If A is an abelian group, then the set Ch(A) of these morphisms forms a group under the operation
- χaχb=χab.
This group is referred to as the character group. 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 a special case of this definition.
[edit] Representation characters
If f is a finite-dimensional representation of a group G, then the character of the representation is the function from G to the complex numbers given by the trace of f. In general, the trace is neither a group homomorphism, nor does the set of traces form a group. The study of representations by means of their characters is called character theory.
[edit] Algebraic characters
If A is an abelian algebra over the complex numbers, a character of A is an algebra homomorphism into the complex numbers. If in addition, A is a *-algebra, then a character is a *-homomorphism into the complex numbers.
