# A note on dual modules and the transpose

## TL;DR

This paper revisits classical matrix algebra results and extends them using module theory to relate transpose and dual representations, applying these ideas to general linear groups over division algebras.

## Contribution

It provides a module-theoretic proof of matrix conjugation to transpose and generalizes the result to central simple algebras with involution, extending representation duality results.

## Key findings

- Matrix conjugation to transpose can be proved via module theory.
- The method extends to central simple algebras with involution.
- Application to GL_n(D) over quaternion division algebra.

## 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 ${\rm GL}_n(F)$ 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 ${\rm GL}_n(D)$ for $D$ a quaternion division algebra over $F$.