Simply Laced Extended Affine Weyl Groups (A Finite Presentation)

Abstract

Extended affine Weyl groups are the Weyl groups of extended affine root systems. Finite presentations for extended affine Weyl groups are known only for nullities $\le 2$, where for nullity $2$ there is only one known such presentation. We give a finite presentation for the class of simply laced extended affine Weyl groups. Our presentation is nullity free if rank $>1$ and for rank $1$ it is given for nullities $\le 3$. The generators and relations are given uniformly for all types, and for a given nullity they can be read from the corresponding finite Cartan matrix and the semilattice involved in the structure of the root system.