Horaires : Le vendredi 21 avril 2017, 15h-16h
Lieu : ENS, salle W
Fields of definition and essential dimension in representation theory
Zinovy Reichstein (UBC, Vancouver)
A classical theorem of Brauer asserts that every finite-dimensional non-modular representation p of a finite group G defined over a field K, whose character takes values in a subfield 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 p is from being definable over k. The classical answer is given by the Schur index of p, which is the smallest degree of a finite field extension l/k such that p 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 p, which measures how far p is from being definable over k in a different way, by using transcendental, rather than algebraic field extensions. This invariant is of interest in both the modular and the non-modular settings. I will also consider the question of which representations of finite groups or finite-dimensional associative algebras have a minimal field of definition with respect to inclusion.