Fields of Definition for Representations of Associative Algebras

Abstract

AbstractWe examine situations, where representations of a finite-dimensionalF-algebraAdefined over a separable extension fieldK/F, have a unique minimal field of definition. Here the base fieldFis assumed to be a field of dimension ≼1. In particular,Fcould be a finite field ork(t) ork((t)), wherekis algebraically closed. We show that a unique minimal field of definition exists if (a)K/Fis an algebraic extension or (b)Ais of finite representation type. Moreover, in these situations the minimal field of definition is a finite extension ofF. This is not the case ifAis of infinite representation type orFfails to be of dimension ≼1. As a consequence, we compute the essential dimension of the functor of representations of a finite group, generalizing a theorem of Karpenko, Pevtsova and the second author.