# 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).

#### References

[1] | H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) MR0000255 Zbl 1024.20502 |

[2] | D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013 |

#### Comments

#### References

[a1] | T.A. Springer, "Invariant theory" , Lect. notes in math. , 585 , Springer (1977) MR0447428 Zbl 0346.20020 |

[a2] | J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) pp. §5.4 MR0323842 Zbl 0254.17004 |

[a3] | Th. Bröcker, T. Tom Dieck, "Representations of compact Lie groups" , Springer (1985) MR0781344 Zbl 0581.22009 |

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Linear representation, invariant of a.

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