- This article refers to the use of the term character theory in mathematics. For the media studies definition, see Character theory (Media). For related senses of the word character, see Character (mathematics).
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.