# 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.

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## Multiplicative character

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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 Template:Harv, 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 are different characters on a group *G* then from it follows that .

## Character of a representation

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The **character of a representation** *φ* of a group *G* on a finite-dimensional vector space *V* over a field *F* is the trace of the representation *φ* Template:Harv. 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.

## See also

- Dirichlet character
- Harish-Chandra character
- Hecke character
- Infinitesimal character
- Alternating character

## References

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## External links

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