Generalized Reflection Root Systems

TL;DR

This paper introduces and classifies generalized reflection root systems (GRRS), extending classical root systems, and provides a comprehensive understanding of their finite and affine cases, including their structure and properties.

Contribution

It defines GRRS, proves classification results for finite and affine types, and describes all such systems, expanding the theory of root systems.

Findings

Finite and affine irreducible GRRSs are fully classified.

All finite GRRSs are described explicitly.

Most affine GRRSs are characterized and classified.

Abstract

We study a combinatorial object, which we call a GRRS (generalized reflection root system); the classical root systems and GRSs introduced by V. Serganova are examples of finite GRRSs. A GRRS is finite if it contains a finite number of vectors and is called affine if it is infinite and has a finite minimal quotient. We prove that an irreducible GRRS containing an isotropic root is either finite or affine; we describe all finite and affine GRRSs and classify them in most of the cases.