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 $E_{6,7,8}.$ 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 $E_{6,7,8}.$ Using our characterization, we enumerate reflectable bases for the finite type E₆.

Keywords:

• Affine reflection systems
• extended affine root systems
• extended affine Weyl groups
• reflectable bases
• minimal generating sets

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B67
• 20F55
• 94C15

Acknowledgement

The first author would like to thank IPM-Isfahan Branch where this research was partially carried out there.