Character (mathematics)

Multiplicative character

Main article: 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 , 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 \chi_1,\chi_2, \ldots , \chi_n are different characters on a group G to the same field then from a_1\chi_1+a_2\chi_2 + \dots + a_n \chi_n = 0 it follows that a_1=a_2=\cdots=a_n=0 .

Character of a representation

Main article: Character theory

The character \chi : G \to F of a representation \phi \colon G\to\mathrm{GL}(V) of a group G on a finite-dimensional vector space V over a field F is the trace of the representation \phi , i.e.

:\chi_\phi(g) = \operatorname{Tr}(\phi(g)) for g \in G

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 1 \times 1 representation in \mathbb{C} (i.e. \mathrm{GL}(V) = \mathrm{GL}(1, \mathbb{C})), the following alternative definition would be equivalent to the above (For abelian groups, every matrix representation decomposes into a direct sum of 1 \times 1 representations. For non-abelian groups, the original definition would be more general than this one):

A character \chi of group (G, \cdot) is a group homomorphism \chi: G \rightarrow \mathbb{C}^* i.e. \chi (x \cdot y)=\chi (x) \chi (y) for all x, y \in G.

If G is a finite abelian group, the characters play the role of harmonics. For infinite abelian groups, the above would be replaced by \chi: G \to \mathbb{T} where \mathbb{T} is the circle group.

References

• Lectures Delivered at the University of Notre Dame

References

1. "character in nLab".