Skip to main content
SpringerLink
Log in

Why Would Multiplicities be Log-Concave?

  • Chapter
The Orbit Method in Geometry and Physics

Part of the book series: Progress in Mathematics ((PM,volume 213))

Abstract

A basic property of entropy in statistical physics is that it is concave as a function of energy. The analog of this in representation theory would be the concavity of the logarithm of the multiplicity of an irreducible representation as a function of its highest weight. We discuss various situations where such concavity can be established or reasonably conjectured and consider some implications of this concavity.

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

Access this chapter

Log in via an institution
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
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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. M. F. Atiyah, Convexity and commuting HamiltoniansBull. London Math. Soc. 14no. 1, (1982), 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. BuchThe saturation conjecture (after A. Knutson and T Tao)math.CO/9810180.

    Google Scholar 

  3. R. Boyer, Infinite traces of AF-algebras and characters of U(oo)J. Operator Theory 9(1983), 205–236.

    MathSciNet  MATH  Google Scholar 

  4. F. BrentiUnimodal log-concave and Polya frequency sequences in combinatoricsMem. AMS 81, no. 413, (1989).

    MathSciNet  Google Scholar 

  5. Yu. Burago and V. ZalgallerGeometric InequalitiesGrundlehren der Mathematischen Wissenschaften 285, Springer-Verlag, Berlin-New York, 1988.

    Google Scholar 

  6. A. Edrei. On the generation function of a doubly infinite, totally positive sequenceTrans. AMS 74 (1953)367–383.

    MathSciNet  MATH  Google Scholar 

  7. W. FultonEigenvalues invariant factors highest weights and Schubert calculus math.AG/9908012, to appear inBull. AMS.

    Google Scholar 

  8. W. GrahamLogarithmic convexity of push forward measuresInvent. Math.123(1996), 315–322.

    MATH  Google Scholar 

  9. V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representationsInvent. Math.67 (1982), no. 3, 515–538.

    Article  MathSciNet  MATH  Google Scholar 

  10. V. Guillemin and S. SternbergConvexity properties of the moment mappingInvent. Math. 67 (1982), no. 3,491–513.

    MathSciNet  MATH  Google Scholar 

  11. V. Guillemin and S. Sternberg, Convexity properties of the moment mapping. IIInvent. Math.77 (1984), no. 3, 533–546.

    Article  MathSciNet  MATH  Google Scholar 

  12. Y. Karshon, Example of a non-log-concave Duistermaat—Heckman measureMath. Res. Lett.3 no. 4, (1996), 537–540.

    MathSciNet  MATH  Google Scholar 

  13. A. Khovanskii, The Newton polytope, the Hilbert polynomial and sums of finite setsFunctional Anal. Appl.26, no. 4, 276–281.

    Google Scholar 

  14. F. Kirwan, Convexity properties of the moment mapping. IIIInvent. Math.77 no. 3, (1984), 547–552.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Knutson and T. Tao, The honeycomb model of GL„(C)tensor products. I. Proof of the saturation conjectureJournal of Amer. Math. Soc.12 no. 4, (1999), 1055–1090.

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Okounkov, Log-Concavity of multiplicities with Application to Characters ofU(oo) Adv. Math.127 no. 2, (1997), 258–282.

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Okounkov, Brunn-Minkowski inequality for multiplicitiesInvent. Math.125 (1996), 405–411.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. OkounkovMultiplicities and Newton polytopesKirillov’s seminar on representation theory, 231–244, AMS Transi. Ser. 2,181, AMS, Providence, RI, 1998.

    MathSciNet  Google Scholar 

  19. A. Okounkov and G. Olshanski, Asymptotics of Jack polynomials as the number of variables goes to infinityInternat. Math. Res. Notices13 (1998), 641–682.

    Article  MathSciNet  Google Scholar 

  20. G. OlshanskiUnitary representations of infinite-dimensional pairs (G K) and the formalism of R. HoweRepresentation of Lie groups and related topics, 269–463, Adv. Stud. Contemp. Math. 7, Gordon and Breach, New York, 1990.

    Google Scholar 

  21. G. Olshanski, On semigroups related to infinite-dimensional groups, Topics in representation theory, 67–101, Adv. Soviet Math. 2, Amer. Math. Soc., Providence, RI, 1991.

    Google Scholar 

  22. R. StanleyLog-concave and unimodal sequences in algebra combinatorics and geometryGraph theory and its applications: East and West (Jinan, 1986), 500–535, Ann. New York Acad. Sci. 576, 1989.

    Google Scholar 

  23. A. Vershik, Description of invariant measures for the action of some infinite-dimensional groupsSoviet Math. Doklady 15(1974), 1396–1400.

    MATH  Google Scholar 

  24. A. Vershik and S. Kerov, Characters and factor representations of the infinite unitary groupSoviet Math. Doklady 26(1982), 570–574.

    MATH  Google Scholar 

  25. D. Voiculescu, Représentations factorielles de type II] de U(oo)J. Math. Pures Appl. 55no. 1, (1976), 1–20.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California at Berkeley, Evans Hall #3840, Berkeley, CA, 94720-3840, USA

    Andrei Okounkov

Authors
  1. Andrei Okounkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Centre de Physique Théorique CNRS, Campus de Luminy, Marseille Cedex 9, F-13288, France

    Christian Duval

  2. Institut Girard Desargues, Université Claude Bernard Lyon 1, Villeurbanne Cedex, F-69622, France

    Valentin Ovsienko

  3. Département des Sciences Mathématiques, Université Montpellier II, Montpelier Cedex 5, F-34095, France

    Laurent Guieu

Additional information

To A. A. Kirillov on the occasion of his 64th birthday

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer Science+Business Media New York

About this chapter

Cite this chapter

Okounkov, A. (2003). Why Would Multiplicities be Log-Concave?. In: Duval, C., Ovsienko, V., Guieu, L. (eds) The Orbit Method in Geometry and Physics. Progress in Mathematics, vol 213. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0029-1_14

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0029-1_14

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6580-1

  • Online ISBN: 978-1-4612-0029-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation