Advertisement

Orthogonal Polynomials and Chevalley Groups

  • Dennis Stanton
Part of the Mathematics and Its Applications book series (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).

Keywords

Finite Group Orthogonal Polynomial Weyl Group Spherical Function Association Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Andrews, G.: 1974, Applications of basic hypergeometric functions, SIAM Rev. 16, 441–484.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Askey, R. and J. Wilson: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, A.M.S. Memoir, to appear.Google Scholar
  3. [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. [4]
    Bailey, W.: 1935, Generalized Hypergeometric Series, Cambridge University Press, Cambridge.zbMATHGoogle Scholar
  5. [5]
    Barmai, E. and T. Ito: Algebraic Combinatorics, Part I, Association Schemes, Benjamin Lecture Notes Series, to appear.Google Scholar
  6. [6]
    Biggs, N.: 1974, Algebraic Graph Theory, Cambridge Tracts in Mathematics, No. 67, Cambridge Univ. Press, Cambridge.Google Scholar
  7. [7]
    Bourbaki, N.: 1968, Groupes et Algebres de Lie, IV, V, V I, Hermann, Paris.Google Scholar
  8. [8]
    Carter, R.: 1972, Simple Groups of Lie Type, Wiley-Interscienc e, London.zbMATHGoogle Scholar
  9. [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. [10]
    Cohen, A. and A. Neumaier: The known distance regular graphs, preprint.Google Scholar
  11. [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. [12]
    Delsarte, P.: 1973, An algebraic approach to the association schemes of coding theory. Philips Res. Repts. Suppl. 10.Google Scholar
  13. [13]
    Delsarte, P.: 1976, Association schemes and t - designs in regular semilattices, J. Comb. Th. A 20, 230–243.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Delsarte, P.: 1978, Bilinear forms over a finite field, with applications to coding theory, J. Comb. Th. A 25, 226–241.MathSciNetGoogle Scholar
  15. [15]
    Delsarte, P.: 1978, Hahn polynomials, discrete harmonics, and t - designs, SIAM J. Appl. Math 34, 157–166.MathSciNetCrossRefGoogle Scholar
  16. [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. [17]
    Delsarte, P. and J.M. Goethals, 1975, Alternating bi-linear forms over GF(q), J. Comb. Th. A 19, 26–50.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Dunkl, C.: 1976, A Krawtchouk polynomial addition theorem and wreath products of symmetric groups, Indiana Univ. Math. J. 25, 335–358.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Dunkl, C.: 1978, An addition theorem for Hahn polynomials: The spherical functions, SIAM J. Math. Anal. 9, 627–637.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Dunkl, C.: 1977, An addition theorem for some q- Hahn polynomials, Monatsh. Math. 85, 5–37.MathSciNetCrossRefGoogle Scholar
  21. [21]
    Dunkl, C.: Spherical function on compact groups and applications to special functions. Symposia Mathematica 22, 145–161.Google Scholar
  22. [22]
    Dunkl, C.: 1981, A difference equation and Hahn polynomials in two variables, Pac. J. Math. 57–71.Google Scholar
  23. [23]
    Dunkl, C.: 1980, Orthogonal polynomials in two variables of q - Hahn and q - Jacobi type, SIAM J. Alg. Disc. Math. 1, 137–151.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    Dunkl, C. and D. Ramirez, 1971, Topics in Harmonic Analysis, Appleton - Century - Crofts, New York.zbMATHGoogle Scholar
  25. [25]
    Egawa, Y.: Association schemes of quadratic forms, to appear.Google Scholar
  26. [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. [27]
    Helgason, S.: 1962, Differential Geometry and Symmetric Spaces, Academic Press, New York.zbMATHGoogle Scholar
  28. [28]
    Higman, D.: 19755, Coherent configurations; Part I: Ordinary representation theory, Geom. Ded. 4, 1–32.Google Scholar
  29. [29]
    Hodges, J.: 1956, Exponential sums for symmetric matrices in a finite field, Arch. Math. 7, 116–121.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [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. [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. [32]
    Koornwinder, T.: 1982, Krawtchouk polynomials, a unification of two different group theoretic interpretations, SIAM J. Math. Anal, 13, 1011–1023.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    Koornwinder, T.: 1981, Clebsch - Gordon coefficients for SU(2) and Hahn polynomials, Math, Centrum Rep, 160.Google Scholar
  34. [34]
    Leonard, D.: 1982, Orthogonal polynomials, duality, and association schemes, SIAM J. Math. Anal. 13, 656–663.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    Miller, W.: 1968, Lie theory and special functions, Academic Press, New York.zbMATHGoogle Scholar
  36. [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. [37]
    Miller, W.: 1970, Lie Theory and q - difference equations, SIAM J. Math. Anal. 1, 171–188.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    Stanton, D.: 1980, Some q - Krawtchouk polynomials on Che-valley groups, Amer, J, Math. 102, 625–662.MathSciNetzbMATHCrossRefGoogle Scholar
  39. [39]
    Stanton, D.: 1980, Product formulas for q - Hahn polynomials, SIAM J. Math. Anal. 11, 100–107.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    Stanton, D.: 1981, Three addition theorems for some q - Krawtchouk polynomials, Geom. Ded. 10, 403–425.zbMATHCrossRefGoogle Scholar
  41. [41]
    Stanton, D.: 1981, A partially ordered set and q-Krawtchouk polynomials, J. Comb. Th. A 30, 276–28k.zbMATHCrossRefGoogle Scholar
  42. [42]
    Stanton, D.: 1983, Generalized n-gons and Chebychev polynomials, J. Comb. Th, A 15–27.Google Scholar
  43. [43]
    Stanton, D,: Harmonics on posets, preprint.Google Scholar
  44. [44]
    Szegö, G.: 1975, Orthogonal Polynomials, Amer. Math. Soc. Colloquium Pub., Vol. 23, Providence.Google Scholar
  45. [45]
    Vere - Jones, D.: 1971, Finite bivariate distributions and semi - groups of non - negative matrices, Quart, J. Math., Oxford (2), 22, 247–270.MathSciNetCrossRefGoogle Scholar
  46. [46]
    Vilenkin, N.: 1968, Special Functions and the Theory of Group Representations, Translations of Amer. Math, Soc. Vol. 22,Google Scholar
  47. [47]
    Vretare, L.: 1976, Elementary spherical functions on symmetric spaces, Math. Scand. 39, 343–358.MathSciNetGoogle Scholar
  48. [48]
    Wang, H.: 1952, Two - point homogeneous spaces, Ann. of Math. 55, 177–191.MathSciNetzbMATHCrossRefGoogle Scholar
  49. [49]
    Wielandt, H.: 1964, Finite Permutation Groups, Academic Press, New York.zbMATHGoogle Scholar

Copyright information

© D. Reidel Publishing Company 1984

Authors and Affiliations

  • Dennis Stanton

There are no affiliations available

Personalised recommendations