A note on dual modules and the transpose

  • Thomas Madsen
  • Alan Roche
  • C. Ryan VinrootEmail author


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.


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

Mathematics Subject Classification

16W10 22E50 15A24 



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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

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