MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I know explicit realization of irreducible representations of simple lie algebra $sl_n$ when the highest weight of that representation is a fundamental weight.Is there any explicit realization of any irreducible representation when the highest weight is not a fundamental weight?

share|cite|improve this question

It depends on how explicit. Abstractly, any finite-dimensional representation is the quotient of a tensor product of fundamental representations, or can be realized as sections of a line bundle over a flag variety (Borel-Weil construction).

If we want really explicit, then we can say more specific things in types ABCDG:

For $sl_n$, see Section 15.3 of the book Representation Theory by Fulton and Harris. For the symplectic Lie algebra and all non-spin representations of the orthogonal Lie algebra, see Section 17.3 and 19.5. The basic spin representations are in Chapter 20, and one can use this and 19.5 to get Weyl-type constructions for the rest of the finite-dimensional representations, see Section 3 of my paper with Weyman for a start on that (the presentation is given, but not worked out in detail). The exceptional types are also discussed in that paper, but only $G_2$ has a complete answer.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.