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.
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Vinroot was supported in part by a grant from the Simons Foundation, Award #280496.
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Madsen, T., Roche, A. & Vinroot, C.R. A note on dual modules and the transpose. Arch. Math. 114, 247–257 (2020). https://doi.org/10.1007/s00013-019-01396-5
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DOI: https://doi.org/10.1007/s00013-019-01396-5
Keywords
- Central simple algebras with involution
- Conjugacy to transpose
- Contragredient representation
- p-adic groups