Reflectable bases for affine reflection systems

TL;DR

This paper introduces the concept of reflectable bases for affine reflection systems, providing a full characterization for tame irreducible types and showing their relation to integral bases in locally finite systems.

Contribution

It defines and characterizes reflectable bases for affine reflection systems, extending the understanding beyond traditional root bases in finite and affine Lie theory.

Findings

Characterization of reflectable bases for tame irreducible affine reflection systems

Exclusion of types E6, E7, E8 from the characterization

Reflectable bases are integral bases in locally finite root systems

Abstract

The notion of a "root base" together with its geometry plays a crucial role in the theory of finite and affine Lie theory. However, it is known that such a notion does not exist for the recent generalizations of finite and affine root systems such as extended affine root systems and affine reflection systems. As an alternative, we introduce the notion of a "reflectable base", a minimal subset $\Pi$ of roots such that the non-isotropic part of the root system can be recovered by reflecting roots of $\Pi$ relative to the hyperplanes determined by $\Pi$. We give a full characterization of reflectable bases for tame irreducible affine reflection systems of reduced types, excluding types $E_{6,7,8}$. As a byproduct of our results, we show that if the root system under consideration is locally finite then any reflectable base is an integral base.