Skip to main content

An Elementary Approach to the Macdonald Identities

  • Conference paper
Book cover q-Series and Partitions

Part of the book series: The IMA Volumes in Mathematics and Its Applications ((IMA,volume 18))

Abstract

Elementary proofs are given for the infinite families of Macdonald identities. The reflections of the Weyl group provide sign-reversing involutions which show that all terms not related to the constant term cancel.

Partially supported by NSF grant DMS-8700995.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Andrews, The Theory of Partitions, Addison-Wesley, Reading, 1976.

    MATH  Google Scholar 

  2. D. Bressoud, Colored tournaments and Weyl’s denominator formula, Eur. J. of Comb., 8 (1987), pp. 245–255.

    MathSciNet  MATH  Google Scholar 

  3. R. Calderbank AND P. Hanlon, The extension to root systems of a theorem on tournaments, J. Comb. Th. A, 41 (1986), pp. 228–245.

    Article  MathSciNet  MATH  Google Scholar 

  4. R. Gustafson, A generalization of Selberg’s integral, research announcement.

    Google Scholar 

  5. R. Gustafson, The Macdonald identities for affine root systems of classical type and hy-pergeometric series very-well-poised on semisimple Lie algebras, preprint.

    Google Scholar 

  6. Laurent Habsieger, MacDonald conjectures and the Selberg integral, in Dennis Stanton (ed.), q-Series and Partitions, IMA Volumes in Mathematics and its Applications, Springer-Verlag, New York, 1989.

    Google Scholar 

  7. I. Macdonald, Affine root systems and Dedekind’s n-function, Inv. Math., 15 (1972), pp. 91–143.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Milne, An elementary proof of the Macdonald identities for A 1 (1), Adv. Math., 57 (1985), pp. 34–70.

    Article  MathSciNet  MATH  Google Scholar 

  9. D. Stanton, Sign variations of the Macdonald identities, SIAM J. Math. Anal., 17 (1986), pp. 1454–1460.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Stembridge, A short proof of Macdonald’s conjecture for the root systems of type A, Proc. AMS, 102 (1988), pp. 777–786.

    MathSciNet  MATH  Google Scholar 

  11. D. Zeilberger, A unified approach to Macdonald’s root-system conjectures, SIAM J. Math. Anal., 19 (1988), pp. 987–1013.

    Article  MathSciNet  MATH  Google Scholar 

  12. D. Zeilberger AND D. Bressoud, A proof of Andrews’ q-Dyson conjecture, Disc. Math., 54 (1985), pp. 201–224.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag New York Inc.

About this paper

Cite this paper

Stanton, D. (1989). An Elementary Approach to the Macdonald Identities. In: Stanton, D. (eds) q-Series and Partitions. The IMA Volumes in Mathematics and Its Applications, vol 18. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-0637-5_11

Download citation

  • DOI: https://doi.org/10.1007/978-1-4684-0637-5_11

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4684-0639-9

  • Online ISBN: 978-1-4684-0637-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics