Powers of Coxeter elements in infinite groups are reduced

الأصل · EN

Let W be an infinite irreducible Coxeter group with (s₁,..., sₙ) the simple generators. We give a simple proof that the word s₁ s₂... sₙ s₁ s₂ >... sₙ... s₁ s₂... sₙ is reduced for any number of repetitions of s₁ s₂ >... sₙ. This result was proved for simply-laced, crystallographic groups by Kleiner and Pelley using methods from the theory of quiver representations. Our proof only using basic facts about Coxeter groups and the geometry of root systems.