Linear representation, invariant of a
A vector $\xi\neq0$ in the space $E$ of a representation $\pi$ of a group $G$ (cf. Representation of a group) such that $\pi(g)\xi=\xi$ for all $g\in G$. An invariant of a linear representation $\pi$ of a Lie algebra $X$ is a vector $\xi\neq0$ in the space $E$ of $\pi$ such that $\pi(x)\xi=0$ for all $x\in X$. In particular, if $\pi$ 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).
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|||D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013|
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|[a3]||Th. Bröcker, T. Tom Dieck, "Representations of compact Lie groups" , Springer (1985) MR0781344 Zbl 0581.22009|
Linear representation, invariant of a. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Linear_representation,_invariant_of_a&oldid=33399