Linear representation

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A homomorphism of a group (respectively an algebra, ring, semi-group) into the group of all invertible linear operators on a vector space (respectively, into the algebra, ring, multiplicative semi-group of all linear operators on ). If is a topological vector space, then a linear representation of on is a representation whose image contains only continuous linear operators on . The space is called the representation space of and the operators , , are called the operators of the representation .


[1] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)



[a1] C.W. Curtis, I. Reiner, "Methods of representation theory" , 1–2 , Wiley (Interscience) (1981–1987)
[a2] J.-P. Serre, "Répresentations linéaires des groupes finis" , Hermann (1967)
[a3] A.I. Shtern, "Theory of group representations" , Springer (1982) (Translated from Russian)
How to Cite This Entry:
Linear representation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article