sp6ℂ and sp2n

  • William Fulton
  • Joe Harris
Part of the Graduate Texts in Mathematics book series (GTM, volume 129)


In the first two sections of this lecture we complete our classification of the representations of the symplectic Lie algebras: we describe in detail the example of\(\mathfrak{s}{\mathfrak{p}_6}\mathbb{C}\)then sketch the representation theory of symplectic Lie algebras in general, in particular proving the existence part of Theorem 14.18 for \(\mathfrak{s}{{\mathfrak{p}}_{{2n}}}\mathbb{C} \). In the final section we describe an analog for the symplectic algebras of the construction given in §15.3 of the irreducible representations of the special linear algebras via Weyl’s construction, though we postpone giving analogous formulas for the decomposition of tensor products of irreducible representations. Sections 17.1 and 17.2 are completely elementary, given the by now standard multilinear algebra of Appendix B. Section 17.3, like §15.3, requires familiarity with the contents of Lecture 6 and Appendix A; but, like that section, it can be skipped without affecting most of the rest of the book.


High Weight Irreducible Representation Weyl Group Symplectic Group High Weight Vector 
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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • William Fulton
    • 1
  • Joe Harris
    • 2
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA

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