Advertisement

Association Schemes with at Most Two Nonlinear Irreducible Characters and Applications to Finite Groups

  • Javad BagherianEmail author
Article
  • 5 Downloads

Abstract

An irreducible character χ of an association scheme is called nonlinear if the multiplicity of χ is greater than 1. The main result of this paper gives a characterization of commutative association schemes with at most two nonlinear irreducible characters. This yields a characterization of finite groups with at most two nonlinear irreducible characters. A class of noncommutative association schemes with at most two nonlinear irreducible character is also given.

Keywords

Association scheme character finite group group-like scheme multiplicity nonlinear 

MR(2010) Subject Classification

05E30 20C15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Antonou, A.: Commutative standard table algebras with at most one nontrivial multiplicity. Comm. Algebra, 43, 2516–2523 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Camina, A. R.: Some conditions which almost characterize Frobenius groups. Israel J. Math., 31, 153–160 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Chillag, D., Macdonald, I. D.: Generalized Frobenius groups. Israel J. Math., 47, 111–122 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Hanaki, A.: Clifford theory for association schemes. J. Algebra, 321, 1686–1695 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Hanaki, A.: Nilpotent schemes and group-like schemes. J. Combin. Theory Ser. A, 115, 226–236 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Hanaki, A.: Characters of association schemes containing a strongly normal closed subset of prime index. Proc. Amer. Math. Soc., 135, 2683–2687 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Hanaki, A.: Character products of association schemes. J. Algebra, 283, 596–603 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Hanaki, A.: Representations of association schemes and their factor schemes. Graphs Combin., 19, 195–201 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Hanaki, A., Hirotsuka, K.: Irreducible representations of wreath products of association schemes. J. Algebraic Combin., 18, 47–52 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Hanaki, A., Miyamoto I.: Classification of association schemes with small number of vertices, published on web: http://math.shinshu-u.ac.jp/˜hanaki/as/Google Scholar
  11. [11]
    Hirasaka, M., Kim K.: On p-covalenced association schemes. J. Combin. Theory Ser. A, 118, 1–8 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Huppert, B.: Character Theory of Finite Groups, Walter de Gruyter, Berlin, New York 1998CrossRefzbMATHGoogle Scholar
  13. [13]
    Isaacs, M.: Character Theory of Finite Groups, Dover, New York, 1994zbMATHGoogle Scholar
  14. [14]
    Muzychuk, M.: A wedge product of association schemes. European J. Combin., 30, 705–715 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Seitz, G.: Finite groups having only one irreducible representation of degree greater than one. Proc. Amer. Math. Soc., 19, 459–461 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Zieschang, P. H.: An Algebraic Approach to Association Schemes, Lecture Notes in Math., vol. 1628, Springer, Berlin, 1996Google Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science (CAS), Chinese Mathematical Society (CAS) and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IsfahanIsfahanIran

Personalised recommendations