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A vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059430/l0594301.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059430/l0594302.png" /> of a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059430/l0594303.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059430/l0594304.png" /> (cf. [[Representation of a group|Representation of a group]]) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059430/l0594305.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059430/l0594306.png" />. An invariant of a linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059430/l0594307.png" /> of a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059430/l0594308.png" /> is a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059430/l0594309.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059430/l05943010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059430/l05943011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059430/l05943012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059430/l05943013.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059430/l05943014.png" /> 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|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|Representation of a Lie algebra]]).
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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|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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) {{MR|0000255}} {{ZBL|1024.20502}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) {{MR|0473097}} {{MR|0473098}} {{ZBL|0228.22013}} </TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top"> H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) {{MR|0000255}} {{ZBL|1024.20502}} </TD></TR>
 
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<TR><TD valign="top">[2]</TD> <TD valign="top"> D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) {{MR|0473097}} {{MR|0473098}} {{ZBL|0228.22013}} </TD></TR>
 
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> T.A. Springer, "Invariant theory" , ''Lect. notes in math.'' , '''585''' , Springer (1977) {{MR|0447428}} {{ZBL|0346.20020}} </TD></TR>
====Comments====
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<TR><TD valign="top">[a2]</TD> <TD valign="top"> J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) pp. §5.4 {{MR|0323842}} {{ZBL|0254.17004}} </TD></TR>
 
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<TR><TD valign="top">[a3]</TD> <TD valign="top"> Th. Bröcker, T. Tom Dieck, "Representations of compact Lie groups" , Springer (1985) {{MR|0781344}} {{ZBL|0581.22009}} </TD></TR>
 
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</table>
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> T.A. Springer, "Invariant theory" , ''Lect. notes in math.'' , '''585''' , Springer (1977) {{MR|0447428}} {{ZBL|0346.20020}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) pp. §5.4 {{MR|0323842}} {{ZBL|0254.17004}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> Th. Bröcker, T. Tom Dieck, "Representations of compact Lie groups" , Springer (1985) {{MR|0781344}} {{ZBL|0581.22009}} </TD></TR></table>
 

Latest revision as of 19:16, 27 October 2023

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
[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
How to Cite This Entry:
Linear representation, invariant of a. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_representation,_invariant_of_a&oldid=24159
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article