G-complete reducibility: Connecting algebra, geometry and representation theory

G-complete reducibility: Connecting algebra, geometry and representation theory

The notion of ‘complete reducibility’ is fundamental in representation theory and commutative algebra. Complete reducibility is responsible for many problems becoming easy to solve, for instance in finite group theory, geometry, Lie groups, Lie algebras and elsewhere.

Translating this concept into purely group-theoretic terms, J.P. Serre extended this notion to ‘G-complete reducibility’ in a reductive algebraic group G. It turns out that this allows many techniques and results from commutative algebra to be extended to the world of algebraic groups, and also has surprising connections to geometry, particularly affine Geometric Invariant Theory. We’ll take a look at these connections and some applications to abstract group theory.

Speaker

Dr Alastair Litterick is a Lecturer in Mathematics in the Department of Mathematical Sciences.