Representation of a group

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.

References

[1]	D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013
[2]	A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) MR0412321 Zbl 0342.22001
[3]	M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018
[4]	D.P. Zhelobenko, A.I. Shtern, "Representations of Lie groups" , Moscow (1981) (In Russian) MR1104272 MR0709598 Zbl 0581.22016 Zbl 0521.22006

References

[a1]	D.J. Benson, "Modular representation theory: New trends and methods" , Lect. notes in math. , 1081 , Springer (1984) MR0765858 Zbl 0564.20004
[a2]	C.W. Curtis, I. Reiner, "Methods of representation theory" , 1–2 , Wiley (Interscience) (1981–1987)
[a3]	W. Feit, "The representation theory of finite groups" , North-Holland (1982) MR0661045 Zbl 0493.20007
[a4]	J.-P. Serre, "Linear representations of finite groups" , Springer (1977) (Translated from French) MR0450380 Zbl 0355.20006
[a5]	B. Huppert, "Endliche Gruppen" , 1 , Springer (1967) pp. 64 MR0224703 Zbl 0217.07201
[a6]	A.W. Knapp, "Representation theory of semisimple groups" , Princeton Univ. Press (1986) MR0855239 Zbl 0604.22001
[a7]	J. Tits, "Tabellen zu den einfachen Lie Grupppen und ihren Darstellungen" , Lect. notes in math. , 40 , Springer (1967)
[a8]	G. Warner, "Harmonic analysis on semisimple Lie groups" , 1–2 , Springer (1972)