In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined (up to isomorphism) by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations.

Applications

Characters of irreducible representations encode many important properties of a group and can thus be used to study its structure. Character theory is an essential tool in the classification of finite simple groups. Close to half of the proof of the Feit–Thompson theorem involves intricate calculations with character values. Easier, but still essential, results that use character theory include Burnside's theorem (a purely group-theoretic proof of Burnside's theorem has since been found, but that proof came over half a century after Burnside's original proof), and a theorem of Richard Brauer and Michio Suzuki stating that a finite simple group cannot have a generalized quaternion group as its Sylow -subgroup.

Definitions

Let be a finite-dimensional vector space over a field and let be a representation of a group on . The character of is the function given by

:<math>\chi_{\rho}(g) = \operatorname{Tr}(\rho(g))</math>

where is the trace.

A character is called irreducible or simple if is an irreducible representation. The degree of the character is the dimension of ; in characteristic zero this is equal to the value . A character of degree 1 is called linear. When is finite and has characteristic zero, the kernel of the character is the normal subgroup:

:<math>\ker \chi_\rho := \left \lbrace g \in G \mid \chi_{\rho}(g) = \chi_{\rho}(1) \right \rbrace, </math>

which is precisely the kernel of the representation . However, the character is not a group homomorphism in general.

Properties

  • Characters are class functions, that is, they each take a constant value on a given conjugacy class. More precisely, the set of irreducible characters of a given group into a field form a basis of the -vector space of all class functions .
  • Isomorphic representations have the same characters. Over a field of characteristic , two representations are isomorphic if and only if they have the same character.
  • If a representation is the direct sum of subrepresentations, then the corresponding character is the sum of the characters of those subrepresentations.
  • If a character of the finite group is restricted to a subgroup , then the result is also a character of .
  • Every character value is a sum of -th roots of unity, where is the degree (that is, the dimension of the associated vector space) of the representation with character and is the order of . In particular, when , every such character value is an algebraic integer.
  • If and is irreducible, then <math display="block">[G:C_G(x)]\frac{\chi(x)}{\chi(1)}</math> is an algebraic integer for all in .
  • If is algebraically closed and does not divide the order of , then the number of irreducible characters of is equal to the number of conjugacy classes of . Furthermore, in this case, the degrees of the irreducible characters are divisors of the order of (and they even divide if ).

Arithmetic properties

Let ρ and σ be representations of . Then the following identities hold:

  • <math>\chi_{\rho \oplus \sigma} = \chi_\rho + \chi_\sigma</math>
  • <math>\chi_{\rho \otimes \sigma} = \chi_\rho \cdot \chi_\sigma</math>
  • <math>\chi_{\rho^*} = \overline {\chi_\rho}</math>
  • <math>\chi_