University of Sydney Algebra Seminar

Ruth Corran (American University of Paris)

Friday 17 March, 12-1pm, Place: Carslaw 375

Root systems for complex reflection groups.

I will speak about joint work with Michel Broué and Jean Michel, motivated by questions coming from the Spetses project. We define a $Z_k$-$rootsystem$ for a complex reflection group on a $k$-vector space $V$, where $Z_k$ is the ring of integers of a number field, $k$. A root is no longer a vector, but something like a rank one $Z_k$-module of $V$. Our definition has natural consequences ; for example, restricting in the obvious way to a parabolic subgroup gives rise to a new root system. In this way, for example, $Z[i]$-root systems naturally arise for Weyl groups of type B ; including one distinct from the Weyl types B and C. We classify root systems (and root and coroot lattices) for complex reflection groups, present Cartan matrices and observe that for spetsial groups, the connection index has a property which generalizes the situation in Weyl groups.