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Are there some good references about representations of classical groups? What are the fundamental representations of classical groups of type $B, D$?

I would like to know explicit formulas of the actions of the Lie algebras ( Lie groups ) on fundamental representations.

Thank you very much.

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  • $\begingroup$ What sort of things are you looking for? One could also approach this more abstractly using induction from the Borel, but that takes a bit more geometric setup. $\endgroup$ Aug 8 '18 at 7:49
  • $\begingroup$ @TobiasKildetoft, thank you very much. I want to know explicit formulas of the actions of the Lie algebras ( Lie groups ) on fundamental representations. $\endgroup$ Aug 8 '18 at 7:52
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This can be found in many places.

On one hand there are general constructions of representations like as quotients of (or generalized) Verma modules or as sections of homogeneous line (or vector) bundles over homogeneous space $G/B$ (or $G/P$).

On the other hand the fundamental representations can be written down explicitly for the classical types and in some cases also for exceptional types.

Let me mention my two favourite books:

  1. Representation Theory: A First Course by William Fulton and Joe Harris
  2. Symmetry, Representations, and Invariants by Roe Goodman and Nolan R. Wallach
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I would add (from memory), for classical groups description

1) Classical groups and geometry by J. Hall

2) The classical groups, their invariants and representations. By H. Weyl

3) (if you read French) La géométrie des groupes classiques by J. Dieudonné

But, if course, I highly recommend Fulton for representations.

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Maybe you like

Tits, Jacques, Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen, Lecture Notes in Mathematics 40. Berlin-Heidelberg-New York: Springer-Verlag VI, 53 S. (1967). ZBL0166.29703.

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Lie groups and Lie algebras by N. Bourbaki, Ch. VIII, § 13 contains, among other things, the detailed descriptions of the fundamental representations.

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