Advertisement

K-theory, genotypes, and biset functors

  • Serge BoucEmail author
Original Article

Abstract

Let p be an odd prime number. In this paper, we show that the genome\(\Gamma (P)\) of a finite p-group P, defined as the direct product of the genotypes of all rational irreducible representations of P, can be recovered from the first group of K-theory \(K_1(\mathbb {Q}P)\). It follows that the assignment \(P\mapsto \Gamma (P)\) is a p-biset functor. We give an explicit formula for the action of bisets on \(\Gamma \), in terms of generalized transfers associated to left free bisets. Finally, we show that \(\Gamma \) is a rational p-biset functor, i.e. that \(\Gamma \) factors through the Roquette category of finite p-groups.

Keywords

K-theory Genotype Whitehead group Biset functor Roquette category transfer 

Mathematics Subject Classification

19B28 20C05 18A99 

Notes

References

  1. 1.
    Barker, L.: Genotypes of irreducible representations of finite \(p\)-groups. J. Algebra 306, 655–681 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bouc, S.: Biset functors for finite groups. In: Lecture Notes in Mathematics, vol. 1990 . Springer, Berlin (2010)Google Scholar
  3. 3.
    Bouc, S.: The Roquette category of finite \(p\)-groups. J. Eur. Math. Soc. 17, 2843–2886 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bouc, S., Romero, N.: The Whitehead group of (almost) extra-special \(p\)-groups with \(p\)-odd. J. Pure Appl. Algebra 223, 86–107 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Curtis, C., Reiner, I.: Methods of representation theory with applications to finite groups and orders. In: Wiley Classics Library, vol. II. Wiley, Hoboken (1990)Google Scholar
  6. 6.
    Curtis, C., Reiner, I.: Methods of representation theory with applications to finite groups and orders. In: Wiley Classics Library, vol. I. Wiley, Hoboken (1990)Google Scholar
  7. 7.
    Oliver, R.: Whitehead groups of finite groups. In: London Mathematical Society Lecture Note Series, vol. 132. Cambridge University Press, Cambridge (1988)Google Scholar
  8. 8.
    Romero, N.: Computing whitehead groups using genetic bases. J. Algebra 450, 646–666 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Roquette, P.: Realisierung von Darstellungen endlicher nilpotenter Gruppen. Arch. Math. 9, 224–250 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rotman, J. J.: An introduction to the theory of groups. Graduate Texts in Mathematics, vol 148, 4th edn. Springer-Verlag, New York (1995)Google Scholar

Copyright information

© Sociedad Matemática Mexicana 2019

Authors and Affiliations

  1. 1.LAMFA-CNRS UMR7352, Université de Picardie Jules VerneAmiens Cedex 01France

Personalised recommendations