Fields of definition for linear representations

Abstract: A classical theorem of Richard Brauer asserts that every finite-dimensional non-modular representation $\rho$ of a finite group $G$ defined over a field $K$, whose character takes values in $k$, descends to $k$, provided that $k$ has suitable roots of unity. If $k$ does not contain these roots of unity, it is natural to ask how far $\rho$ is from being definable over $k$. The classical answer to this question is given by the Schur index of $\rho$, which is the smallest degree of a finite field extension $l/k$ such that $\rho$ can be defined over $l$. In this talk, based on joint work with Nikita Karpenko, Julia Pevtsova and Dave Benson, I will discuss another invariant, the essential dimension of $\rho$, which measures ''how far'' $\rho$ is from being definable over $k$ in a different way by using transcendental, rather than algebraic field extensions. I will also talk about recent results of Federico Scavia on essential dimension of representations of algebras.

group theory

Audience: researchers in the topic

Finite Groups in Valencia

Series comments: The seminar meets weekly from February 26th to March 30th 2021.

For a detailed schedule and registration details, please visit the seminar webpage at

sites.google.com/view/finite-groups-seminar2021