Character theory

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
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:
which is precisely the kernel of the representation . However, the character is "not" a group homomorphism in general.
Properties.
Arithmetic properties.
Let ρ and σ be representations of . Then the following identities hold: