Weyl Groups Associated with Affine Reflection Systems of Type $A_1$ (Coxeter Type Defining Relations)

TL;DR

This paper provides the first non-trivial presentations of Weyl groups associated with affine reflection systems of type A1, including hyperbolic and extended affine Weyl groups, offering a foundational model for broader types.

Contribution

It introduces novel presentations for Weyl groups of affine reflection systems of type A1, including hyperbolic and extended affine cases, in a general setting.

Findings

Presented a new presentation for Weyl groups of type A1

Derived a finite presentation for extended affine Weyl groups of type A1

Provided a model for presentations of other types

Abstract

In this paper, we offer a presentation for the Weyl group of an affine reflection system $R$ of type $A_1$ as well as a presentation for the so called hyperbolic Weyl group associated with an affine reflection system of type $A_1$. Applying these presentations to extended affine Weyl groups, and using a description of the center of the hyperbolic Weyl group, we also give a new finite presentation for an extended affine Weyl group of type $A_1$. Our presentation for the (hyperbolic) Weyl group of an affine reflection system of type $A_1$ is the first non-trivial presentation given in such a generality, and can be considered as a model for other types.