Linear representation theory

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 $G$ over a field $k$ is a homomorphism $\rho :G\to GL(V)$ where $V$ is a vector space over $k$ and $GL(V)$ denotes the general linear group of $V$, viz the group of automorphisms of $V$ as a $k$-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

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Results

The following is a list of useful results in linear representation theory:

Important theorems

• 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

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.