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 language | English |
|---|---|
| Pages (from-to) | 291-304 |
| Number of pages | 14 |
| Journal | Proceedings of the Edinburgh Mathematical Society |
| Volume | 62 |
| Issue number | 1 |
| Early online date | 22 Nov 2018 |
| DOIs | |
| Publication status | Published - Feb 2019 |
Bibliographical note
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.
Keywords
- Modular representation
- field of definition
- finite representation type
- essential dimension
- 2010 Mathematics subject classification: Primary 16G10 16G60 20C05
- modular representation