TY - JOUR

T1 - Fields of Definition for Representations of Associative Algebras

AU - Benson, Dave

AU - Reichstein, Zinovy

N1 - The authors are grateful to Mathieu Florence, Roberto Pirisi and Julia Pevtsova for helpful comments. The research of the first author was supported by the Collaborative Research Group in Geometric and Cohomological Methods in Algebra at the Pacific Institute for the Mathematical Sciences, Vancouver, Canada (2016).
The second author was partially supported by NSERC Discovery Grant 250217-2012.

PY - 2019/2

Y1 - 2019/2

N2 - 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.

AB - 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.

KW - Modular representation

KW - field of definition

KW - finite representation type

KW - essential dimension

KW - 2010 Mathematics subject classification: Primary 16G10 16G60 20C05

KW - modular representation

UR - http://www.scopus.com/inward/record.url?scp=85057206508&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/fields-definition-representations-associative-algebras

U2 - 10.1017/S0013091518000391

DO - 10.1017/S0013091518000391

M3 - Article

VL - 62

SP - 291

EP - 304

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

SN - 0013-0915

IS - 1

ER -