Linear representation theory: Difference between revisions
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==Important definitions== | ==Important definitions== | ||
{{further|[[Basic definitions in linear representation theory]]}} | |||
* [[linear representation]] | * [[linear representation]] | ||
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* [[irreducible linear representation]] | * [[irreducible linear representation]] | ||
* [[completely reducible linear representation]] | * [[completely reducible linear representation]] | ||
* [[character]] | |||
* [[Character table]] | |||
* [[Tensor product of linear representations]] | |||
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The following is a list of useful results in linear representation theory: | The following is a list of useful results in linear representation theory: | ||
===Important theorems=== | |||
* [[Maschke's theorem]] | * [[Maschke's theorem]] | ||
* [[Schur's lemma]] | |||
* [[Character orthogonality theorem]] | |||
===Other important results=== | |||
* [[Irreducible complex representation of abelian group is one dimensional]] | |||
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==Applications== | ==Applications== | ||
===In pure mathematics=== | |||
Representation theory has numerous applications in pure mathematics, since we can transform problems in abstract algebra to problems in linear algebra. For example, many results on this wiki that are stated purely in the language of group theory can be proven using representation theory, e.g. [[Burnside’s theorem]]. | |||
===In applied mathematics and the natural sciences=== | |||
When symmetry arises in nature, groups, and thus their representations, are a useful mathematical concept for applied mathematicians and scientists. For example, chemists or quantum physicists may study the symmetries of molecules or particles respectively, which can be made into a problem within the representation theory of groups. | |||
Latest revision as of 19:17, 10 November 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
Further information: Basic definitions in linear representation theory
- linear representation
- subrepresentation
- irreducible linear representation
- completely reducible linear representation
- character
- Character table
- Tensor product of linear representations
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
Results
The following is a list of useful results in linear representation theory:
Important theorems
Other important results
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
Applications
In pure mathematics
Representation theory has numerous applications in pure mathematics, since we can transform problems in abstract algebra to problems in linear algebra. For example, many results on this wiki that are stated purely in the language of group theory can be proven using representation theory, e.g. Burnside’s theorem.
In applied mathematics and the natural sciences
When symmetry arises in nature, groups, and thus their representations, are a useful mathematical concept for applied mathematicians and scientists. For example, chemists or quantum physicists may study the symmetries of molecules or particles respectively, which can be made into a problem within the representation theory of groups.