## 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 BrouĂ© and Jean Michel, motivated by questions coming from the Spetses project. We define a \(Z_k\)-\(root\; system\) 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.