Editing Group representation (section)

{{Short description|Group homomorphism into the general linear group over a vector space}}
{{Distinguish|Presentation of a group}}

[[File:Hexagon_Reflections.png|thumb|right|250px|A representation of a [[group (mathematics)|group]] "acts" on an object. A simple example is how the [[Dihedral group|symmetries of a regular polygon]], consisting of reflections and rotations, transform the polygon.]]

In the [[mathematics|mathematical]] field of [[representation theory]], '''group representations''' describe abstract [[group (mathematics)|groups]] in terms of [[bijective]] [[linear transformation]]s of a [[vector space]] to itself (i.e. vector space [[automorphism]]s); in particular, they can be used to represent group elements as [[invertible matrix|invertible matrices]] so that the group operation can be represented by [[matrix multiplication]].

In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules.

Representations of groups allow many [[group theory|group-theoretic]] problems to be reduced to problems in [[linear algebra]]. In [[physics]], they describe how the [[symmetry group]] of a physical system affects the solutions of equations describing that system.

The term ''representation of a group'' is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a [[homomorphism]] from the group to the [[automorphism group]] of an object. If the object is a vector space we have a ''linear representation''. Some people use ''realization'' for the general notion and reserve the term ''representation'' for the special case of linear representations. The bulk of this article describes linear representation theory; see the last section for generalizations.