Linear representation theory: Difference between revisions
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The '''linear representation theory''' of groups (or '''representation theory''' or '''group representation theory''') is the study of [[linear representations]] of [[groups]]. A linear representation of a group <math>G</math> over a [[field]] <math>k</math> is a [[homomorphism of groups|homomorphism]] <math>\rho:G \to GL(V)</math> where <math>V</math> is a [[vector space]] over <math>k</math> and <math>GL(V)</math> denotes the [[general linear group]] of <math>V</math>, viz the group of automorphisms of <math>V</math> as a <math>k</math>-vector space. | The '''linear representation theory''' of groups (or '''representation theory''' or '''group representation theory''') is the study of [[linear representations]] of [[groups]]. A linear representation of a group <math>G</math> over a [[field]] <math>k</math> is a [[homomorphism of groups|homomorphism]] <math>\rho:G \to GL(V)</math> where <math>V</math> is a [[vector space]] over <math>k</math> and <math>GL(V)</math> denotes the [[general linear group]] of <math>V</math>, viz the group of automorphisms of <math>V</math> as a <math>k</math>-vector space. | ||
==Important definitions== | |||
* [[linear representation]] | |||
* [[subrepresentation]] | |||
* [[irreducible linear representation]] | |||
* [[completely reducible linear representation]] | |||
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==Results== | ==Results== | ||
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* [[Maschke's theorem]] | * [[Maschke's theorem]] | ||
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==Applications== | |||
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Revision as of 17:49, 23 October 2023
The linear representation theory of groups (or representation theory or group representation theory) is the study of linear representations of groups. A linear representation of a group over a field is a homomorphism where is a vector space over and denotes the general linear group of , viz the group of automorphisms of as a -vector space.
Important definitions
- linear representation
- subrepresentation
- irreducible linear representation
- completely reducible linear representation
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Results
The following is a list of useful results in linear representation theory:
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Applications
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