Advertisement

A note on dual modules and the transpose

  • Thomas Madsen
  • Alan Roche
  • C. Ryan VinrootEmail author
Article
  • 11 Downloads

Abstract

It is a classical result in matrix algebra that any square matrix over a field can be conjugated to its transpose by a symmetric matrix. For F a non-Archimedean local field, Tupan used this to give an elementary proof that transpose inverse takes each irreducible smooth representation of \({\mathrm{GL}}_n(F)\) to its dual. We re-prove the matrix result and related observations using module-theoretic arguments. In addition, we write down a generalization that applies to central simple algebras with an involution of the first kind. We use this generalization to extend Tupan’s method of argument to \({\mathrm{GL}}_n(D)\) for D a quaternion division algebra over F.

Keywords

Central simple algebras with involution Conjugacy to transpose Contragredient representation p-adic groups 

Mathematics Subject Classification

16W10 22E50 15A24 

Notes

References

  1. 1.
    Gelfand, I.M., Kazhdan, D.A.: Representations of the group \({\rm GL}(n,K)\) where \(K\) is a local field. Lie groups and their representations (Proceedings of Summer School, Bolyai János Bolyai Mathematical Society, Budapest, 1971), pp. 95–118. Halsted, New York (1975)Google Scholar
  2. 2.
    Kaplansky, I.: Linear Algebra and Geometry: A Second Course, 2nd edn. Chelsea Publishing Company, New York (1974)zbMATHGoogle Scholar
  3. 3.
    Knus, M.-A., Merkurjev, A., Rost, M., Tignol, J.-P.: The Book of Involutions. AMS Colloquium Publications, vol. 44. American Mathematical Society, Providence (1998)zbMATHGoogle Scholar
  4. 4.
    Muić, G., Savin, G.: Complementary series for Hermitian quaternionic groups. Can. Math. Bull. 43(1), 90–99 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Raghuram, A.: On representations of \(p\)-adic \({{\rm GL}}_2(D)\). Pac. J. Math. 206(2), 451–464 (2002)Google Scholar
  6. 6.
    Roche, A., Vinroot, C.R.: Dualizing involutions for classical and similitude groups over local non-Archimedean fields. J. Lie Theory 27(2), 419–434 (2017)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Roche, A., Vinroot, C.R.: A factorization result for classical and similitude groups. Can. Math. Bull. 61(1), 174–190 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Taussky, O., Zassenhaus, H.: On the similarity transformation between a matrix and its transpose. Pac. J. Math. 9, 893–896 (1959)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Tupan, A.: A triangulation of \({{\rm GL}}(n, F )\). Represent. Theory 10, 158–163 (2006)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsYoungstown State UniversityYoungstownUSA
  2. 2.Department of MathematicsUniversity of OklahomaNormanUSA
  3. 3.Department of MathematicsCollege of William and MaryWilliamsburgUSA

Personalised recommendations