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

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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

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