Representation of a group
2010 Mathematics Subject Classification: Primary: 20B Secondary: 22F05 [MSN][ZBL]
A homomorphism of the group into the group of all invertible transformations of a set $V$.
A permutation representation is a homomorphism to the symmetric group $S_V$: a group action of $G$ on $V$: cf. Permutation group.
A representation $\rho$ of a group $G$ is called linear if $V$ is a vector space over a field $k$ and if the transformations $\rho(g)$, $g\in G$, are linear. Often, linear representations are, for shortness, simply termed representations (cf. Representation theory). In the theory of representations of abstract groups the theory of finite-dimensional representations of finite groups is best developed (cf. Finite group, representation of a; Representation of the symmetric groups).
If $G$ is a topological group, then one considers continuous linear representations of $G$ on a topological vector space $V$ (cf. Continuous representation; Representation of a topological group). If $G$ is a Lie group and $V$ is a finite-dimensional space over $\mathbf R$ or $\mathbf C$, then a continuous linear representation is automatically real analytic. Analytic and differentiable representations of a Lie group are defined also in the infinite-dimensional case (cf. Analytic representation; Infinite-dimensional representation). To each differentiable representation $\rho$ of a Lie group $G$ corresponds some linear representation of its Lie algebra — the differential representation of $\rho$ (cf. Representation of a Lie algebra). If $G$ is moreover connected, then its finite-dimensional representations are completely determined by their differentials. The most developed branch of the representation theory of topological groups is the theory of finite-dimensional linear representations of semi-simple Lie groups, which is often formulated in the language of Lie algebras (cf. Finite-dimensional representation; Representation of the classical groups; Cartan theorem on the highest weight vector), the representation theory of compact groups, and the theory of unitary representations (cf. Representation of a compact group; Unitary representation).
For algebraic groups one has the theory of rational representations (cf. Rational representation), which is in many aspects analogous to the theory of finite-dimensional representations of Lie groups.
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Representation of a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Representation_of_a_group&oldid=35282