S-Rings, Gelfand Pairs and Association Schemes

  • Kenneth W. Johnson
Part of the Lecture Notes in Mathematics book series (LNM, volume 2233)


The construction by Frobenius of group characters described in Chap.  1 may be generalized to the case where a permutation group G acting on a finite set has the Gelfand pair property explained below. A general setting which encompasses and extends this is that of an association scheme. For any association scheme there is available a character theory, which in the case where the scheme arises from a group coincides with that of group characters. The development of the theory is described in this chapter. Firstly Schur investigated centralizer rings of permutation groups, then Wielandt defined S-rings over a group. The theory of association schemes provides a character theory even when a group is not present, and this can be applied to obtain a character theory for a loop or quasigroup. In particular a Frobenius reciprocity result was obtained for quasigroup characters, and it was realized that this is available for arbitrary association schemes. A further idea, that of fusion of characters of association schemes, leads to interesting results including a “magic rectangle” condition.


  1. 9.
    R.A. Bailey, Association Schemes: Designed Experiments, Algebra and Combinatorics. Cambridge Studies in Advanced Mathematics, vol. 84 (Cambridge University Press, Cambridge, 2004)Google Scholar
  2. 10.
    E. Bannai, T. Ito, Algebraic Combinatorics I: Association Schemes (Benjamin/Cummings, London, 1984)zbMATHGoogle Scholar
  3. 11.
    E. Bannai, S.-Y. Song, The character tables of Paige’s simple Moufang loops and their relationship to the character tables of PSL(2,q). Proc. Lond. Math. Soc. 58, 209–236 (1989)MathSciNetCrossRefGoogle Scholar
  4. 20.
    W. Blaschke, G. Bol, Geometrie der Gewebe. Topologische Fragen der Differentialgeometrie (J.W. Edwards, Ann Arbor, 1944)Google Scholar
  5. 21.
    H.I. Blau, Table algebras. Eur. J. Comb. 30(6), 1426–1455 (2009)MathSciNetCrossRefGoogle Scholar
  6. 32.
    A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance-regular Graphs. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 3 (Springer, Berlin, 1989), p. 18Google Scholar
  7. 34.
    R.H. Bruck, A Survey of Binary Systems (Springer, Berlin, 1958)CrossRefGoogle Scholar
  8. 39.
    W. Burnside, The Theory of Groups of Finite Order, 2nd edn. (Cambridge University Press, Cambridge, 1911)zbMATHGoogle Scholar
  9. 41.
    P.J. Cameron, in Oligomorphic Permutation Groups. London Mathematical Society Lecture Notes, vol. 152 (Cambridge University Press, Cambridge 1996)Google Scholar
  10. 42.
    P.J. Cameron, Coherent configurations, association schemes and permutation groups, in Groups, Combinatorics & Geometry (Durham, 2001) (World Scientific, River Edge, 2003), pp. 55–71zbMATHGoogle Scholar
  11. 43.
    P.J. Cameron, Aspects of infinite permutation groups, in Groups St. Andrews 2005, I. London Mathematical Society Lecture Notes vol. 339 (Cambridge University Press, Cambridge, 2007), pp. 1–35Google Scholar
  12. 44.
    P.J. Cameron, K.W. Johnson, An investigation of countable B-groups. Math. Proc. Camb. Philos. Soc. 102, 223–231 (1987)MathSciNetCrossRefGoogle Scholar
  13. 45.
    P.J. Cameron, J.H. van Lint, Designs, Graphs, Codes and Their Links. London Mathematical Society Student Texts, vol. 22 (Cambridge University Press, Cambridge, 1991)Google Scholar
  14. 46.
    P.J. Cameron, P.M. Neumann, D.N. Teague, On the degrees of primitive permutation groups. Math. Z. 180, 141–149 (1982)MathSciNetCrossRefGoogle Scholar
  15. 47.
    P.J. Cameron, J.M. Goethals, J.J. Seidel, The Krein condition, spherical designs, Norton algebras and permutation groups. Indag. Math. 40, 196–206 (1987)zbMATHGoogle Scholar
  16. 55.
    A. Chan, C.D. Godsil, A. Munemasa, Four-weight spin models and Jones pairs. Trans. Am. Math. Soc. 355, 2305–2325 (2003)MathSciNetCrossRefGoogle Scholar
  17. 76.
    P. Delsarte, An algebraic approach to the Association scheme of coding theory. Philips Research Reports: Supplements, No. 10 (1973)Google Scholar
  18. 98.
    S, Evdokimov, I. Ponomarenko, Schurity of S-rings over a cyclic group and the generalized wreath product of permutation groups (Russian). Algebra Anal. 24(3), 84–127 (2012); translation in St. Petersburg Math. J. 24(3), 431–460 (2013)Google Scholar
  19. 103.
    J.S. Frame, The double cosets of a finite group. Bull. Am. Math. Soc. 47, 458–467 (1941)MathSciNetCrossRefGoogle Scholar
  20. 104.
    J.S. Frame, Double coset matrices and group characters. Bull. Am. Math. Soc. 49, 81–92 (1943)MathSciNetCrossRefGoogle Scholar
  21. 106.
    G. Frobenius, Über Gruppencharaktere (Sitzungsber. Preuss. Akad. Wiss, Berlin, 1896), pp. 985–1021; Ges Abh. III, pp. 1–37Google Scholar
  22. 116.
    F.R. Gantmacher, The Theory of Matrices, vol. I (Chelsea Publishing, White River Junction, 1959)zbMATHGoogle Scholar
  23. 123.
    C.D. Godsil, Algebraic Combinatorics. Chapman and Hall Mathematics Series (Chapman & Hall, New York, 1993)zbMATHGoogle Scholar
  24. 133.
    R. Haggkvist, J.C.M. Janssen, All-even latin squares. Discrete Math. 157, 199–206 (1996)MathSciNetCrossRefGoogle Scholar
  25. 166.
    K.W. Johnson, S-rings over loops, right mapping groups and transversals in permutation groups. Math. Proc. Camb. Philos. Soc. 89, 433–443 (1981)MathSciNetCrossRefGoogle Scholar
  26. 167.
    K.W. Johnson, Loop transversals and the centralizer ring of a permutation group. Math. Proc. Camb. Philos. Soc. 94, 411–416 (1983)MathSciNetCrossRefGoogle Scholar
  27. 170.
    K.W. Johnson, Sharp Characters of quasigroups. Eur. J. Comb. 14, 103–112 (1993)MathSciNetCrossRefGoogle Scholar
  28. 175.
    K.W. Johnson, J.D.H. Smith, Characters of finite quasigroups. Eur. J. Comb. 5, 43–50 (1984)MathSciNetCrossRefGoogle Scholar
  29. 176.
    K.W. Johnson, J.D.H. Smith, Characters of finite quasigroups II. Ind uced characters. Eur. J. Comb. 7, 131–137 (1986)MathSciNetCrossRefGoogle Scholar
  30. 177.
    K.W. Johnson, J.D.H. Smith, A note on character induction in association schemes. Eur. J. Comb. 7, 139 (1986)MathSciNetCrossRefGoogle Scholar
  31. 178.
    K.W. Johnson, J.D.H. Smith, Characters of finite quasigroups. III. Quotients and fusion. Eur. J. Comb. 10, 47–56 (1989)MathSciNetCrossRefGoogle Scholar
  32. 179.
    K.W. Johnson, J.D.H. Smith, Characters of finite quasigroups V, Linear characters. Eur. J. Comb. 10, 449–456 (1989)MathSciNetCrossRefGoogle Scholar
  33. 183.
    K.W. Johnson, S.Y. Song, J.D.H. Smith, Characters of finite quasigroups, VI. Critical examples and doubletons. Eur. J. Comb. 11, 267–275 (1990)zbMATHGoogle Scholar
  34. 190.
    A. Kerber, Applied Finite Group Actions, 2nd edn. (Springer, Berlin, 1999)CrossRefGoogle Scholar
  35. 197.
    W. Knapp, On Burnside’s method. J. Algebra 175, 644–660 (1995)MathSciNetCrossRefGoogle Scholar
  36. 206.
    T. Luczak, L. Pyber, On random generation of the symmetric group. Comb. Probab. Comput. 2, 505–512 (1993)MathSciNetCrossRefGoogle Scholar
  37. 209.
    G.W. Mackey, The Scope and History of Commutative and Noncommutative Harmonic Analysis. History of Mathematics, vol. 5 (American Mathematical Society, Providence; London Mathematical Society, London, 1992)Google Scholar
  38. 221.
    M. Muzychuk, M. Klin, R. Pöschel, The isomorphism problem for circulant graphs via Schur ring theory, in Codes and Association Schemes (Piscataway, 1999). DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, vol. 56 (American Mathematical Society, Providence, 2001), pp. 241–264CrossRefGoogle Scholar
  39. 222.
    G.P. Nagy, P. Vojtěchovský, LOOPS: computing with quasigroups and loops in GAP, version 2.2.0.
  40. 223.
    P.M. Neumann, The context of Burnside’s contribution to group theory, in The Collected Papers of William Burnside, ed. by P.M. Neumann, A.J.S. Mann, J.C. Thompson (Oxford University Press, Oxford, 2004), 15–37Google Scholar
  41. 224.
    M. Niemenmaa, T. Kepka, On connected transversals to abelian subgroups in finite groups. Bull. Lond. Math. Soc. 24, 343–346 (1992)MathSciNetCrossRefGoogle Scholar
  42. 238.
    C. Praeger, Quasiprimitivity: structure and combinatorial applications. Discret. Math. 264, 211–224 (2003)MathSciNetCrossRefGoogle Scholar
  43. 254.
    I. Schur, Neuer Beweis eines Satzes von W. Burnside. Jahresber. Deutsch. Math.-Verein 17, 171–176 (1908); Ges. Abh I, 266–271Google Scholar
  44. 255.
    I. Schur, Zur Theorie der einfach transitiven Permutationsgruppen. Sitzungsber. Preuss. Akad. Wiss. Phys-Math Klasse 598–623 (1933); Ges. Abh III, 266–291Google Scholar
  45. 257.
    W.R. Scott, Group Theory (Prentice Hall, Englewood Cliffs, 1964)Google Scholar
  46. 259.
    J.D.H. Smith, Centraliser rings of multiplication groups of quasigroups. Math. Proc. Camb. Philos. Soc. 79, 427–431 (1976)MathSciNetCrossRefGoogle Scholar
  47. 261.
    J.D.H. Smith, Combinatorial characters of quasigroups, in Coding Theory and Design Theory, Part I: Coding Theory, ed. by D. Ray-Chaudhuri (Springer, New York, 163–187, 1990)CrossRefGoogle Scholar
  48. 263.
    J.D.H. Smith, An Introduction to Quasigroups and Their Representations (Chapman and Hall, London, 2006)CrossRefGoogle Scholar
  49. 268.
    D. Stanton, Orthogonal polynomials and Chevalley groups, in Special Functions: Group Theoretical Aspects and Applications. Mathematics and Its Application (Reidel, Dordrecht, 1984), pp. 87–128CrossRefGoogle Scholar
  50. 271.
    O. Tamaschke, Schur-Ringe, (Vorlesungen an der Univ. Tübingen 1969) (Bibliographisches Institut, Mannheim, 1970)Google Scholar
  51. 293.
    H. Wielandt, Finite Permutation Groups (Academic, New York, 1964)zbMATHGoogle Scholar
  52. 294.
    H. Wielandt, Mathematical Works, Volume 1: Group Theory (Walter de Gruyter, Berlin, New York, 1994)zbMATHGoogle Scholar
  53. 297.
    P.-H. Zieschang, An Algebraic Approach to Association Schemes. Springer Lecture Notes in Mathematics, vol. 1626 (Springer, Berlin, 1996)Google Scholar

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Authors and Affiliations

  • Kenneth W. Johnson
    • 1
  1. 1.Department of MathematicsPennsylvania State UniversityAbingtonUSA

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