Linear representation, invariant of a

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A vector in the space of a representation of a group (cf. Representation of a group) such that for all . An invariant of a linear representation of a Lie algebra is a vector in the space of such that for all . In particular, if is a representation of a linear group in a space of multilinear functions, the given definition of the invariant of a linear representation coincides with the classical definition. The invariants of a linear representation arising from restricting an irreducible representation to a subgroup play an important role in the representation theory of Lie groups and Lie algebras (cf. Representation of a Lie algebra).


[1] H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946)
[2] D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)



[a1] T.A. Springer, "Invariant theory" , Lect. notes in math. , 585 , Springer (1977)
[a2] J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) pp. §5.4
[a3] Th. Bröcker, T. Tom Dieck, "Representations of compact Lie groups" , Springer (1985)
How to Cite This Entry:
Linear representation, invariant of a. Encyclopedia of Mathematics. URL:,_invariant_of_a&oldid=12505
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article