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Difference between revisions of "Linear representation, invariant of a"

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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Weyl,   "The classical groups, their invariants and representations" , Princeton Univ. Press (1946)</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)</TD></TR></table>
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<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><TR><TD valign="top">[a1]</TD> <TD valign="top"> T.A. Springer,   "Invariant theory" , ''Lect. notes in math.'' , '''585''' , Springer (1977)</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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> Th. Bröcker,   T. Tom Dieck,   "Representations of compact Lie groups" , Springer (1985)</TD></TR></table>
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<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>

Revision as of 10:54, 1 April 2012

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).

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
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=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