Root systems constructed by folding of the extended Dynkin diagrams

TL;DR

This paper introduces a method to construct finite root systems by folding extended Dynkin diagrams using automorphisms induced by the stabilizer subgroup of the extended affine Weyl group.

Contribution

It presents a novel approach to generate finite root systems through folding of extended Dynkin diagrams based on automorphisms from the stabilizer subgroup.

Findings

Constructed finite root systems via folding of extended Dynkin diagrams.

Identified automorphism subgroups of the extended Dynkin diagrams.

Provided a new perspective on the structure of root systems.

Abstract

The extended affine Weyl group of a root system is the semidirect product of the corresponding Weyl group by its coweight lattice. The stabilizer subgroup of the extended affine Weyl group with respect to the corresponding fundamental alcove induces a subgroup of automorphisms of the extended Dynkin diagram. In this paper, we construct a finite root system by folding by the elements of the subgroup.