Fields of Definition for Representations of Associative Algebras

Dave Benson (Corresponding Author), Zinovy Reichstein

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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.
Original languageEnglish
Pages (from-to)291-304
Number of pages14
JournalProceedings of the Edinburgh Mathematical Society
Volume62
Issue number1
Early online date22 Nov 2018
DOIs
Publication statusPublished - Feb 2019

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Keywords

  • Modular representation
  • field of definition
  • finite representation type
  • essential dimension
  • 2010 Mathematics subject classification: Primary 16G10 16G60 20C05
  • modular representation

ASJC Scopus subject areas

  • Mathematics(all)

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