# Fields of definition for representations of associative algebras

## TL;DR

This paper investigates conditions under which representations of finite-dimensional algebras over certain fields have a unique minimal field of definition, and computes the essential dimension of the representation functor for finite groups.

## Contribution

It establishes criteria for the existence and finiteness of minimal fields of definition for algebra representations over $C_1$-fields, extending previous results to new classes of algebras and fields.

## Key findings

- Unique minimal field of definition exists if extension is algebraic or algebra is of finite type.
- Minimal fields of definition are finite extensions of the base field in these cases.
- Computes the essential dimension of the representation functor for finite groups.

## 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 $C_1$-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.