University of Sydney Algebra Seminar

Jean Michel (Université Paris 7)

Friday 5th December, 12.05-12.55pm, Carslaw 373 ***NOTE UNUSUAL ROOM***

Automorphisms of complex reflection groups

I report on joint work with Ivan Marin. We prove that any automorphism of an irreducible finite complex reflection group is, apart from the single exception of the symmetric group on six letters, the product of automorphisms of 3 kinds:

• a 'diagram' automorphism, induced by the ambient linear group;
• a central automorphism;
• a 'Galois' automorphism, a new kind of automorphisms which come from an embedding of the Galois group of the defining field into the outer automorphism group using Galois cohomology properties of the reflection representation.

We then extend the result to a large class of non-irreducible groups using an extension of the Krull-Remak-Schmidt theorem.