Submission: 2017, Feb 21
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 C_1-field.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 C_1. As a consequence, we compute the essential dimension of the functor of representations of a finite group, generalizing a theorem of N. Karpenko, J. Pevtsova and the second author.
2010 Mathematics Subject Classification: 16G10, 16G60, 20C05
Keywords and Phrases: Modular representation, field of definition, finite representation type, essential dimension
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