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Abstract: A classical theorem of Brauer asserts that every finite-dimensional
non-modular representation Ï 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 Ï is from being
definable over k. The classical answer is given by the Schur index of
Ï, which is the smallest degree of a finite field extension l/k
such that Ï 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 Ï. This
invariant measures "how far" Ï is from being definable over k in a
different way, by using transcendental, rather than algebraic field
extensions. I will also discuss related work on representations of
algebras, due to Federico Scavia.