Symmetric Systems and their Applications to Root Systems Extended by Abelian Groups

TL;DR

This paper studies root systems extended by abelian groups, introducing symmetric systems to analyze their Weyl groups and group homomorphisms, revealing structural properties and kernel isomorphisms.

Contribution

It introduces the concept of symmetric systems and establishes kernel isomorphisms in the context of extended root systems by abelian groups.

Findings

Kernel of U to W homomorphism is isomorphic to the kernel of abelianized U to abelianized W

Symmetric systems provide a discrete framework for analyzing extended root systems

Structural relations between Weyl groups and conjugation presentations are clarified

Abstract

We investigate the class of root systems R obtained by extending an irreducible root system by a torsion-free group G. In this context there is a Weyl group W and a group U with the presentation by conjugation. We show under additional hypotheses that the kernel of the natural homomorphism from U to W is isomorphic to the kernel of the homomorphism from the abelianization of U to that of W. For this we introduce the concept of a symmetric system, a discrete version of the concept of a symmetric space. Mathematics Subject Classification 2000: 20F55, 17B65, 17B67, 22E65, 22E40. Key Words and Phrases: Weyl group, root system, presentation by conjugation, extended affine Weyl group (EAWeG), extended affine root system (EARS), irreducible root system extended by an abelian group.