## University of Sydney Algebra Seminar

# Don Taylor (University of Sydney)

## Friday 31st October, 12.05-12.55pm, Carslaw 159

### The structure of finite unitary reflection groups

Given a finite dimensional Hilbert space *V*, a
(complex or unitary) reflection
is a linear transformation of *V*, of finite order,
which fixes a hyperplane pointwise.

In 1954 Shephard and Todd obtained the complete classification of finite groups generated by reflections. For the primitive groups they relied on the classification of finite groups generated by homologies, which was achieved by Blichfeldt, Mitchell, and others, 50 years before.

In this talk I will outline a proof of this theorem based on the
classification of embeddings between line systems, where a line system in *V*
is a set of one-dimensional subspaces of *V* with prescribed angles between
lines. This approach leads to the identification of the reflection groups as
linear groups over finite fields.