On simply-laced generalized root systems

TL;DR

This paper proves the existence and uniqueness of the Euler form in simply-laced generalized root systems, classifies their isomorphism classes via admissible diagrams, and links Coxeter elements to Carter's admissibility.

Contribution

It establishes foundational properties of Euler forms and classifies simply-laced generalized root systems using Coxeter elements and Carter's admissible diagrams.

Findings

Uniqueness and existence of Euler form for simply-laced systems

Classification of systems via admissible diagrams

Connection between Coxeter elements and Carter's admissibility

Abstract

We show the uniqueness and existence of the Euler form for a simply-laced generalized root system. This enables us to show that the Coxeter element for a simply-laced generalized root system is admissible in the sense of R.~W.~Carter. As an application, the isomorphism classes of simply-laced generalized root systems with positive definite Cartan forms are classified by Cartar's admissible diagrams associated to their Coxeter elements.