Proof of Existence and Uniqueness of Simple Root Systems, Clarified
William Ian Holt · OpenSIUC (Southern Illinois University Carbondale) · 2020
Humphreys (1990) defines a simple root system for a finite reflection group which is an important concept fundamental to the understanding of reflection groups as well as Coxeter complexes and Coxeter groups. The proof that Humphreys uses to establish the existence and uniqueness of these systems follows an indirect method that left portions of the proof as exercises to the reader. I present a more complete and direct proof using different organization and methods.