The Weyl group of type $A_1$ root systems extended by an abelian group

TL;DR

This paper explores the structure of Weyl groups extended by abelian groups for type A_1 root systems, establishing conditions for their isomorphism based on 2-independence in associated subsets.

Contribution

It introduces a criterion for when the Weyl group and the conjugation group are isomorphic, based on 2-independence of certain subsets in the extended root system.

Findings

The homomorphism from the conjugation group to the Weyl group is an isomorphism if and only if a specific subset is 2-independent.

The 2-independence condition is characterized by linear independence over the Galois field F_2.

The results connect algebraic properties of the extended root system with the structure of associated reflection groups.

Abstract

We investigate the class of root systems $R$ obtained by extending an $A_1$-type irreducible root system by a free abelian group $G$. In this context there is a Weyl group $W$ and a group $U$ with the presentation by conjugation. Both groups are reflection groups with respect to a discrete symmetric space $T$ associated to $R$. We show that the natural homomorphism $U\to W$ is an isomorphism if and only if an associated subset $T^{ab}\setminus\{0\}$ of $G_2=G/2G$ is 2-independent, i.e. its image under the map $G_2\to G_2\otimes G_2, g\mapsto g\otimes g$ is linearly independent over the Galois field $F_2$.