Skip to main content
SpringerLink
Log in
Book cover

Asymptotic Combinatorics with Applications to Mathematical Physics pp 201–221Cite as

Algebraic length and Poincaré series on reflection groups with applications to representations theory

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1815))

Abstract

Let W be a reflection group generated by a finite set of simple reflections S. We determine suficient and necessary condition for invertibility and positive definitness of the Poincaré series\( \sum\nolimits_w q ^{\ell (w)} w \) , where ℓ(w)}denotes the algebraic length on W relative to S. Generalized Poincaré series are defined and similar results for them are proved.

In case of finite W, representations are constructed which are canonically associated with the algebraic length.For crystallographic groups (Weyl groups)these representations are decomposed into irreducible components.Positive definitness of certain functions involving generalized lengths on W is proved.The proofs don’t make use of the classification of finite reflection groups.Examples are provided.

Partially supported by KBN (Poland)under grant 2 P03A 054 15.

Partially supported by KBN (Poland)under grant 2 P03A 048 15.

This is a preview of subscription content, 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   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Berg, C., Christensen, J.P.: Harmonic Analysis on Semigroups:Theory of Positive Definite Functions.Graduate Texts in Mathematics,100, Springer Verlag, New York,(1984)

    Google Scholar 

  2. Bourbaki, N.: Groupes et algèbres de Lie, Ch.4-6. Hermann, Paris, (1968)

    Google Scholar 

  3. Bożejko, M.:Positive and negative kernels on discrete groups.Lectures at Heidelberg University,(1987)

    Google Scholar 

  4. Bożejko, M., Januszkiewicz, T., Spatzier, R.J.: Infinite Coxeter groups do not have Kazhdan’s property. J.Operator Theory,19, 63–68 (1988)

    MathSciNet  MATH  Google Scholar 

  5. Bożejko, M., Picardello, M.A.:Weakly amenable groups and amalgamated products. Proc.Amer.Math.Soc., 117, 1039–1046 (1993)

    Google Scholar 

  6. Bożejko, M., Speicher, R.:An example of a generalized Brownian motion. Comm.Math.Phys.,137, 519–531 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bożejko, M., Speicher, R.:Completely positive maps on Coxeter groups,deformed commutation relations and operator spaces.Math.Ann.,300, 97–120(1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bożejko, M., Speicher, R.:Interpolation between bosonic and fermionic relations given by generalized Brownian motions.Math.Z.,222,135–160 (1996)

    MathSciNet  Google Scholar 

  9. Bożejko, M., Kümerer, B., Speicher, R.: q-Gaussian processes:nonclassical and classical aspects.Comm.Math.Phys.(to appear)

    Google Scholar 

  10. Carter, R.: Simple Groups of Lie Type.Wiley-Interscience, London (1972)

    MATH  Google Scholar 

  11. Coxeter, H.S.M.:Regular Complex Polytopes.Cambridge University Press, Cambridge,(1974)

    Google Scholar 

  12. Coxeter, H.S.M.:Discrete groups generated by reflections.Annals of Math., 35, 588–621 (1934)

    Google Scholar 

  13. Coxeter, H.S.M., Moser, W.O.:Generators and Relations for Discrete Groups, 2nd ed. Springer-Verlag, New York,(1965)

    MATH  Google Scholar 

  14. de Canniere, J., Haagerup, U.:Multipliers of the Fourier algebra of some simple Lie groups and their discrete subgroups. Amer.Math.Soc.,107, 455–500 (1983)

    Google Scholar 

  15. de la Harpe, P.:Groupes de Coxeter infinites non affines.Expo.Math.,5, 91–96(1987)

    MATH  Google Scholar 

  16. Dykema, K., Nica, A.: On the Fock representation of the q-commutation relations. J.Reine Angew.Math.,440, 201–212 (1993)

    MATH  MathSciNet  Google Scholar 

  17. Greenberg, O.W.: Particles with small violations of Fermi or Bose statistics. Phys.Rev.D,43, 4111–4120 (1991)

    Article  MathSciNet  Google Scholar 

  18. Grove, L.C., Benson, C.T.:Finite Reflection Groups, 2nd ed.Springer-Verlag, New York (1985)

    MATH  Google Scholar 

  19. Hiller, H.:Geometry of Coxeter Groups. Pitman, London,(1982)

    Google Scholar 

  20. Humphreys, J.E.:Reflection groups and Coxeter groups.Cambridge University Press, Cambridge,(1990)

    Google Scholar 

  21. Januszkiewicz, T.: For right angled Coxeter groups z ∣g∣ is a coeficient of uniformly bounded representation.Proc.Amer.Math.Soc.,119,1115–1119 (1993)

    MATH  MathSciNet  Google Scholar 

  22. Jorgensen, P.E.T., Schmitt, L M., Werner, R.F.:q-canonical commutation relations and stability of the Cuntz algebra.Pacific Math.J.,165,131–151 (1994)

    MathSciNet  Google Scholar 

  23. Pytlik, T., Szwarc, R.: An analytic series of uniformly bounded representations of free groups. Acta Math.,157, 287–309 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  24. Szwarc, R.:Groups acting on trees and approximation properties of the Fourier algebra, J.Funct.Anal.,95, 320–343 (1991)

    Google Scholar 

  25. Szwarc, R.: Structure of geodesics in the Cayley graph of infinite Coxeter groups. Preprint (1996)

    Google Scholar 

  26. Tits, J.:Buildings of Spherical Type and Finite BN-pairs.Lecture Notes in Math.386, Springer-Verlag,(1974)

    Google Scholar 

  27. Valette, A.:Weak amenability of right-angled Coxeter groups.Proc.Amer. Math.Soc.,119, 1331–1334 (1993)

    Google Scholar 

  28. Zagier, D.: Realizabilty of a model in infinite statistics.Comm.Math.Phys., 147,199–210 (1992)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Wroclaw University, pl.Grunwaldzki 2/4, 50-384, Wroclaw, Poland

    Marek Bożejko & Ryszard Szwarc

Authors
  1. Marek Bożejko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ryszard Szwarc
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. St. Petersburg Department of the Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191011, St. Petersburg, Russia

    Anatoly M. Vershik

  2. St. Petersburg Department of the Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191011, St. Petersburg, Russia

    Yuri Yakubovich

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Bożejko, M., Szwarc, R. (2003). Algebraic length and Poincaré series on reflection groups with applications to representations theory. In: Vershik, A.M., Yakubovich, Y. (eds) Asymptotic Combinatorics with Applications to Mathematical Physics. Lecture Notes in Mathematics(), vol 1815. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44890-X_9

Download citation

  • DOI: https://doi.org/10.1007/3-540-44890-X_9

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40312-8

  • Online ISBN: 978-3-540-44890-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation