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Fields of Definition for Representations of Associative Algebras

Part of: Representation theory of groups Representation theory of rings and algebras

Published online by Cambridge University Press:  22 November 2018

Dave Benson  and
Zinovy Reichstein
Dave Benson
Affiliation:
Institute of Mathematics, University of Aberdeen, King's College, Aberdeen AB24 3UE, UK ([email protected])
Zinovy Reichstein
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, Canada ([email protected])
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Abstract

We examine situations, where representations of a finite-dimensional F-algebra A defined over a separable extension field K/F, have a unique minimal field of definition. Here the base field F is assumed to be a field of dimension ≼1. In particular, F could be a finite field or k(t) or k((t)), where k is algebraically closed. We show that a unique minimal field of definition exists if (a) K/F is an algebraic extension or (b) A is of finite representation type. Moreover, in these situations the minimal field of definition is a finite extension of F. This is not the case if A is of infinite representation type or F fails 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.

Keywords

modular representationfield of definitionfinite representation typeessential dimension

MSC classification

Primary: 16G10: Representations of Artinian rings
Secondary: 16G60: Representation type (finite, tame, wild, etc.) 20C05: Group rings of finite groups and their modules
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 291 - 304
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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