Editing Character theory (section)

{{Short description|Concept in mathematical group theory}}
{{About|the use of the term character theory in mathematics|related senses of the word character|Character (mathematics)}}
In [[mathematics]], more specifically in [[group theory]], the '''character''' of a [[group representation]] is a [[function (mathematics)|function]] on the [[group (mathematics)|group]] that associates to each group element the [[trace (linear algebra)|trace]] of the corresponding [[matrix (mathematics)|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 number|complex]] representation of a [[finite group]] is determined (up to [[isomorphism]]) by its character. The situation with representations over a [[field (mathematics)|field]] of positive [[characteristic (algebra)|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 representation theory|modular representations]].