Characterization and enumeration of reflectable bases of types E6, 7, 8

Abstract

We characterize reflectable bases for affine reflection systems (extended affine root systems) of types E-6,E-7,E-8. This completes the characterization of reflectable bases for reduced affine reflections systems. We prove that a subset of non-isotropic roots is a reflectable base if and only if it is a minimal generating set for the root lattice. Our approach also provides a new less involved proof for simply laced types different from E-6,E- 7,E- 8. Using our characterization, we enumerate reflectable bases for the finite type E-6.