Skip to main content
SpringerLink
Log in

Orthogonal Polynomials and Chevalley Groups

  • Chapter
Special Functions: Group Theoretical Aspects and Applications

Part of the book series: Mathematics and Its Applications ((MAIA,volume 18))

Abstract

This paper is a survey of recent work on orthogonal polynomials and Chevalley groups. The orthogonal polynomials we emphasize are those given by basic hypergeometric series, or q-series. The group theoretic significance of these polynomials is that they are the spherical functions for Chevalley groups over the finite field GF(q).

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
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. Andrews, G.: 1974, Applications of basic hypergeometric functions, SIAM Rev. 16, 441–484.

    Article  MathSciNet  MATH  Google Scholar 

  2. Askey, R. and J. Wilson: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, A.M.S. Memoir, to appear.

    Google Scholar 

  3. Askey, R. and J. Wilson: A set of orthogonal polynomials that generalize the Racah coefficients or 6 - j symbols, SIAM J. Math. Anal. 10, 1008–10l6.

    Google Scholar 

  4. Bailey, W.: 1935, Generalized Hypergeometric Series, Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  5. Barmai, E. and T. Ito: Algebraic Combinatorics, Part I, Association Schemes, Benjamin Lecture Notes Series, to appear.

    Google Scholar 

  6. Biggs, N.: 1974, Algebraic Graph Theory, Cambridge Tracts in Mathematics, No. 67, Cambridge Univ. Press, Cambridge.

    Google Scholar 

  7. Bourbaki, N.: 1968, Groupes et Algebres de Lie, IV, V, V I, Hermann, Paris.

    Google Scholar 

  8. Carter, R.: 1972, Simple Groups of Lie Type, Wiley-Interscienc e, London.

    MATH  Google Scholar 

  9. Cartier, P.: 19733 ‘Harmonic analysis on trees’, in C.C. Moore, (ed.), Harmonic Analysis on Homogeneous Spaces, Proc. Symp. Pure Math., No. 26, Amer. Math. Soc., Providence, pp. 419–424.

    Google Scholar 

  10. Cohen, A. and A. Neumaier: The known distance regular graphs, preprint.

    Google Scholar 

  11. Curtis, C., N. Iwahori and R. Kilmoyer: Hecke algebras and characters of parabolic type of finite groups with (B,N) pairs.

    Google Scholar 

  12. Delsarte, P.: 1973, An algebraic approach to the association schemes of coding theory. Philips Res. Repts. Suppl. 10.

    Google Scholar 

  13. Delsarte, P.: 1976, Association schemes and t - designs in regular semilattices, J. Comb. Th. A 20, 230–243.

    Article  MathSciNet  MATH  Google Scholar 

  14. Delsarte, P.: 1978, Bilinear forms over a finite field, with applications to coding theory, J. Comb. Th. A 25, 226–241.

    MathSciNet  Google Scholar 

  15. Delsarte, P.: 1978, Hahn polynomials, discrete harmonics, and t - designs, SIAM J. Appl. Math 34, 157–166.

    Article  MathSciNet  Google Scholar 

  16. Delsarte, P.: 1976, Properties and application of the recurrence F(i + l, k + l, n + l) = qk+l F(i, k+l, n)- qk F(i, k, n), SIAM J. Appl. Math, 31, 262–270.

    Google Scholar 

  17. Delsarte, P. and J.M. Goethals, 1975, Alternating bi-linear forms over GF(q), J. Comb. Th. A 19, 26–50.

    Article  MathSciNet  MATH  Google Scholar 

  18. Dunkl, C.: 1976, A Krawtchouk polynomial addition theorem and wreath products of symmetric groups, Indiana Univ. Math. J. 25, 335–358.

    Article  MathSciNet  MATH  Google Scholar 

  19. Dunkl, C.: 1978, An addition theorem for Hahn polynomials: The spherical functions, SIAM J. Math. Anal. 9, 627–637.

    Article  MathSciNet  MATH  Google Scholar 

  20. Dunkl, C.: 1977, An addition theorem for some q- Hahn polynomials, Monatsh. Math. 85, 5–37.

    Article  MathSciNet  Google Scholar 

  21. Dunkl, C.: Spherical function on compact groups and applications to special functions. Symposia Mathematica 22, 145–161.

    Google Scholar 

  22. Dunkl, C.: 1981, A difference equation and Hahn polynomials in two variables, Pac. J. Math. 57–71.

    Google Scholar 

  23. Dunkl, C.: 1980, Orthogonal polynomials in two variables of q - Hahn and q - Jacobi type, SIAM J. Alg. Disc. Math. 1, 137–151.

    Article  MathSciNet  MATH  Google Scholar 

  24. Dunkl, C. and D. Ramirez, 1971, Topics in Harmonic Analysis, Appleton - Century - Crofts, New York.

    MATH  Google Scholar 

  25. Egawa, Y.: Association schemes of quadratic forms, to appear.

    Google Scholar 

  26. Gasper, G.: 1975, ‘Positivity and special function’, in R. Askey (ed.), Theory and Application of Special Functions, Academic Press, New York, 375–433.

    Google Scholar 

  27. Helgason, S.: 1962, Differential Geometry and Symmetric Spaces, Academic Press, New York.

    MATH  Google Scholar 

  28. Higman, D.: 19755, Coherent configurations; Part I: Ordinary representation theory, Geom. Ded. 4, 1–32.

    Google Scholar 

  29. Hodges, J.: 1956, Exponential sums for symmetric matrices in a finite field, Arch. Math. 7, 116–121.

    Article  MathSciNet  MATH  Google Scholar 

  30. James, G. and A. Kerber, 1981, ‘The Representation Theory of the Symmetric Group’, in G.C. Rota (ed.), Encylopedia of Mathematics and its Applications, Vol 16, Addison - Wesley, Reading, Mass.

    Google Scholar 

  31. Koornwinder, T.: 1975, ‘Two variable analogues of the classical orthogonal polynomials’, in R. Askey (ed.), Theory and Application of Special Function, Academic Press, New York, 435–495.

    Google Scholar 

  32. Koornwinder, T.: 1982, Krawtchouk polynomials, a unification of two different group theoretic interpretations, SIAM J. Math. Anal, 13, 1011–1023.

    Article  MathSciNet  MATH  Google Scholar 

  33. Koornwinder, T.: 1981, Clebsch - Gordon coefficients for SU(2) and Hahn polynomials, Math, Centrum Rep, 160.

    Google Scholar 

  34. Leonard, D.: 1982, Orthogonal polynomials, duality, and association schemes, SIAM J. Math. Anal. 13, 656–663.

    Article  MathSciNet  MATH  Google Scholar 

  35. Miller, W.: 1968, Lie theory and special functions, Academic Press, New York.

    MATH  Google Scholar 

  36. Miller, W.: 1977, ‘Symmetry and Separation of Variables’, in G.C. Rota (ed.), Encyclopedia of Mathematics and its Applications, Vol. Addison - Wesley, Reading, Mass.

    Google Scholar 

  37. Miller, W.: 1970, Lie Theory and q - difference equations, SIAM J. Math. Anal. 1, 171–188.

    Article  MathSciNet  MATH  Google Scholar 

  38. Stanton, D.: 1980, Some q - Krawtchouk polynomials on Che-valley groups, Amer, J, Math. 102, 625–662.

    Article  MathSciNet  MATH  Google Scholar 

  39. Stanton, D.: 1980, Product formulas for q - Hahn polynomials, SIAM J. Math. Anal. 11, 100–107.

    Article  MathSciNet  MATH  Google Scholar 

  40. Stanton, D.: 1981, Three addition theorems for some q - Krawtchouk polynomials, Geom. Ded. 10, 403–425.

    Article  MATH  Google Scholar 

  41. Stanton, D.: 1981, A partially ordered set and q-Krawtchouk polynomials, J. Comb. Th. A 30, 276–28k.

    Article  MATH  Google Scholar 

  42. Stanton, D.: 1983, Generalized n-gons and Chebychev polynomials, J. Comb. Th, A 15–27.

    Google Scholar 

  43. Stanton, D,: Harmonics on posets, preprint.

    Google Scholar 

  44. Szegö, G.: 1975, Orthogonal Polynomials, Amer. Math. Soc. Colloquium Pub., Vol. 23, Providence.

    Google Scholar 

  45. Vere - Jones, D.: 1971, Finite bivariate distributions and semi - groups of non - negative matrices, Quart, J. Math., Oxford (2), 22, 247–270.

    Article  MathSciNet  Google Scholar 

  46. Vilenkin, N.: 1968, Special Functions and the Theory of Group Representations, Translations of Amer. Math, Soc. Vol. 22,

    Google Scholar 

  47. Vretare, L.: 1976, Elementary spherical functions on symmetric spaces, Math. Scand. 39, 343–358.

    MathSciNet  Google Scholar 

  48. Wang, H.: 1952, Two - point homogeneous spaces, Ann. of Math. 55, 177–191.

    Article  MathSciNet  MATH  Google Scholar 

  49. Wielandt, H.: 1964, Finite Permutation Groups, Academic Press, New York.

    MATH  Google Scholar 

Download references

Authors
  1. Dennis Stanton
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Mathematics Department, University of Wisconsin, Madison, USA

    R. A. Askey

  2. Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

    T. H. Koornwinder

  3. Department of Mathematics, University of Siegen, Germany

    W. Schempp

Rights and permissions

Reprints and permissions

Copyright information

© 1984 D. Reidel Publishing Company

About this chapter

Cite this chapter

Stanton, D. (1984). Orthogonal Polynomials and Chevalley Groups. In: Askey, R.A., Koornwinder, T.H., Schempp, W. (eds) Special Functions: Group Theoretical Aspects and Applications. Mathematics and Its Applications, vol 18. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-9787-1_2

Download citation

  • DOI: https://doi.org/10.1007/978-94-010-9787-1_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-0319-6

  • Online ISBN: 978-94-010-9787-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation