Advertisement

Combinatorial Characters of Quasigroups

  • Jonathan D. H. Smith
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 20)

Abstract

Over a century ago, when the character theory of finite abelian groups had become established, Dedekind began the programme of extending the theory to finite non-abelian groups. Having made little headway a decade later, he proposed the task to Frobenius. Developments progressed rapidly in Frobenius’ hands, along both Dedekind’s original group determinant approach and Frobenius’ new approach that is now considered part of the theory of association schemes. Shortly afterwards, representation theory methods using matrices took over, and have dominated ever since.

Keywords

Conjugacy Class Association Scheme Character Table Steiner Triple System Character Theory 
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. [BI]
    E. Bannai and T. Ito, Algebraic Combinatorics I, Benjamin/Cummings, Menlo Park 1984.zbMATHGoogle Scholar
  2. [BS]
    A. Barlotti and K. Strambach, The geometry of binary systems, Adv. in Math. 49 (1983), 1–105.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [B1]
    R.H. Bruck, Contributions to the theory of loops, Trans. Amer. Math. Soc. 60 (1946), 245–354.MathSciNetzbMATHGoogle Scholar
  4. [B2]
    R.H. BRUCK, A Survey of Binary Systems, Springer-Verlag, Berlin 1958.zbMATHGoogle Scholar
  5. [B3]
    R.H. Bruck, What is a loop?, in Studies in Modern Algebra (A.A. Albert ed.), M.A.A. Studies in Mathematics No. 2, Prentice Hall, Enlewood Cliffs 1963.Google Scholar
  6. [CG]
    P.J. Cameron, J.M. Goethals, and J.J. Seidel, The Krein condition, spherical designs, Norton algebras and permutation groups, Indag. Math. 81 (1978), 196–206.MathSciNetGoogle Scholar
  7. [Ch]
    O. Chein, H. Pflugfelder and J.D.H. Smith (eds.), Theory and Applications of Quasigroups and Loops, Heldermann Verlag, Berlin 1989.Google Scholar
  8. [Co]
    J.H. Conway, A simple construction for the Fischer-Griess monster group, Inv. Math. 79 (1985), 513–540.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [Cr]
    C.W. Curtis and I. Reiner, Methods of Representation Theory, Vol. I, Wiley, New York 1981.zbMATHGoogle Scholar
  10. [De]
    P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Repts. Supp. 10 (1973).Google Scholar
  11. [Dg]
    P. Diaconis, Group Representations in Probability and Statistics, Institute of Mathematical Statistics Lecture notes-Monograph Series Vol. 11, Institute of Mathematical Statistics, Hayward California, (1988).zbMATHGoogle Scholar
  12. [Di]
    P.G.L.-Dirichlet (R. Dedekind ed.), Vorlesungen über Zahlentheorie, (3rd. ed.), Braunschweig 1879.Google Scholar
  13. [Do]
    J.L. Doob, Stochastic Processes, Wiley, New York 1953.zbMATHGoogle Scholar
  14. [Fr]
    F.G. Frobenius (J.-P. Serre ed.), Gesammelte Abhandlungen Bd. III, Springer, Berlin 1968.Google Scholar
  15. [Ga]
    C.F. Gauss (A.A. Clarke Tr.), Disquisitiones Arithmeticae, Yale University Press, New Haven 1966.zbMATHGoogle Scholar
  16. [H1]
    T. Hawkings, The origin of the theory of group characters, Archive Hist. Exact Sci. 7 (1971), 142–170.CrossRefGoogle Scholar
  17. [H2]
    T. Hawkins, New light on Frobenius’ creation of the theory of group characters, Archive Hist. Exact Sci. 12 (1974), 217–243.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [He]
    H. Heyer, Convolution semigroups of probability measures on Gelfand pairs, Expo. Math. 1 (1983), 3–45.MathSciNetzbMATHGoogle Scholar
  19. [Hu]
    B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin 1967.zbMATHGoogle Scholar
  20. [J1]
    K.W. Johnson, S-rings over loops, right mapping groups and transversals in permutation groups, Math. Proc. Comb. Phil. Soc. 89 (1981), 433–422.zbMATHCrossRefGoogle Scholar
  21. [J2]
    K.W. Johnson, Latin square determinants, in proceedings of the 1986 Montréal conference on Extremal Set Theory and Relational Structures (I.G. Rosenberg ed.), Cambridge University Press, Cambridge 1988.Google Scholar
  22. [J3]
    K.W. Johnson and J.D.H. Smith, Characters of finite quasigroups, Europ. J. Combinatorics 5 (1984), 43–50.MathSciNetzbMATHGoogle Scholar
  23. [J4]
    K.W. Johnson and J.D.H. Smith, Characters of finite quasigroups II: induced characters, Europ. J. Combinatorics 7 (1986), 131–137.MathSciNetzbMATHGoogle Scholar
  24. [J5]
    K.W. Johnson and J.D.H. Smith, A note on character induction in association schemes, Europ. J. Combinatorics 7 (1986), 139.MathSciNetGoogle Scholar
  25. [J6]
    K.W. Johnson and J.D.H. Smith, Characters of finite quasigroups III: quotients and fusion, Europ. J. Combinatorics, 10 (1989), 47–56.MathSciNetzbMATHGoogle Scholar
  26. [J7]
    K.W. Johnson and J.D.H. Smith, Characters of finite quasigroups IV: products and superschemes, Europ. J. Combinatorics, to appear.Google Scholar
  27. [J8]
    K.W. Johnson and J.D.H. Smith, Characters of finite quasigroups V: linear characters, IMA preprint series # 413 (1988).Google Scholar
  28. [Ma]
    I.G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford 1979.zbMATHGoogle Scholar
  29. [Ml]
    A.I. Mal’cev, On the general theory of algebraic systems (Russian), Mat. Sb. N.S. 35 (77) (1954), 3–20.MathSciNetGoogle Scholar
  30. [Mo]
    T. Molien, Ueber die Invarianten der Linearen Substitutionsgruppen, Sitzungsber. d. Akad. d. Wiss. Berlin 1897, 1152–1156.Google Scholar
  31. [Pi]
    G. Pickert, Projektive Ebene, Springer-Verlag, Berlin 1975.Google Scholar
  32. [RS]
    A.B. Romanowska and J.D.H. Smith, Modal Theory-An Algebraic Approach to Order Geometry, and Convexity, Heldermann Verlag, Berlin 1985.zbMATHGoogle Scholar
  33. [Se]
    J.-P. Serre (L.L. Scott Tr.), Linear Representations of Finite Groups, Springer-Verlag, New York 1977.zbMATHGoogle Scholar
  34. [S1]
    J.D.H. Smith, Centrality, Ph.D. Thesis (unpublished), Cambridge University, 1974.Google Scholar
  35. [S2]
    J.D.H. Smith, Centraliser rings of multiplication groups on quasigroups, Math. Proc. Camb. Phil. Soc. 79 (1976), 427–431.zbMATHCrossRefGoogle Scholar
  36. [S3]
    J.D.H. Smith, Mal’cev Varieties, Springer Lecture Notes in Mathematics No. 554, Springer-Verlag, Berlin 1976.zbMATHGoogle Scholar
  37. [S4]
    J.D.H. Smith, Representation Theory of Infinite Groups and Finite Quasigroups, Séminaire de Mathématiques Supérieures No. 101, Université de Montréal, Montréal 1986.zbMATHGoogle Scholar
  38. [S5]
    J.D.H. Smith, Quasigroups, association schemes, and Laplace operators on almost-periodic functions, in proceedings of the 1986 Montréal conference on Extremal Set Theory and Relational Structures (I.G. Rosenberg ed.), Cambridge University Press, Cambridge 1988.Google Scholar
  39. [S6]
    J.D.H. Smith, Entropy, character theory and centrality of finite quasigroups, IMA preprint series #416 (1988).Google Scholar
  40. [S7]
    J.D.H. Smith, Induced class functions are conditional expectations, preprint.Google Scholar
  41. [So]
    S.Y. Song, The Character Tables of Certain Association Schemes, Ph.D. Thesis (unpublished), Ohio State University, 1987.Google Scholar
  42. [Ta]
    O. Tamaschke, S-Ringe und verallgemeinerte Charaktere auf endlichen Gruppen, Math. Zeitschr. 84 (1964), 101–119.MathSciNetzbMATHCrossRefGoogle Scholar
  43. [W1]
    H. Weber, Beweis des Satzes dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen fähig ist, Math. Ann. 20 (1882), 301–329.MathSciNetCrossRefGoogle Scholar
  44. [W2]
    H. Weber, Theorie der Abel’schen Zahlkïrper, Acta Math. 8 (1886), 193–263.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [W3]
    H. Weber, Theorie der Abel’schen Zahlkörper, Acta Math. 9 (1887), 105–130.MathSciNetCrossRefGoogle Scholar
  46. [W4]
    H. Weber, Lehrbuch der Algebra, Bd. 2, Braunschweig 1896.zbMATHGoogle Scholar
  47. [Wi]
    H. Wielandt, Finite Permutation Groups, Academic Press, New York 1964.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1990

Authors and Affiliations

  • Jonathan D. H. Smith
    • 1
  1. 1.Department of MathematicsIowa State UniversityAmesUSA

Personalised recommendations