Powers of Coxeter Elements in Infinite Groups Are Reduced

Let W be an infinite irreducible Coxeter group with (s1, . . . , sn) the simple generators. We give a simple proof that the word s1s2 · · · sns1s2 · · · sn · · · s1s2 · · · sn is reduced for any number of repetitions of s1s2 · · · sn. This result was proved for simplylaced, 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. Let W be a Coxeter group with S the generating set of reflections. An element c ∈ W of the form s1 · · · sn, with s1, . . . , sn some ordering of the elements of S, is called a Coxeter element. It is a result of Howlett [2] that, if W is infinite, then any Coxeter element has infinite order. In [8], it is shown that, in each classical affine group, there is an ordering (s1, . . . , sn) of the simple generators such that the word s1 · · · sns1 · · · sn · · · s1 · · · sn is reduced for any number of repetitions of s1 · · · sn. 1 Fomin and Zelevinsky [1, Corollary 9.6] proved a version of this result for Coxeter groups with bipartite diagrams; they show that, if S = I ⊔ J is a partition of S into two sets so that all the elements in each set commute, and if W is irreducible and infinite then the word ∏ i∈I si ∏ j∈J sj ∏ i∈I si ∏ j∈J sj · · · ∏ i∈I si ∏ j∈J sj is reduced for any number of repetitions of ∏ i∈I si ∏ j∈J sj. Recently, Kleiner and Pelley [5], relying heavily on results of Kleiner and Tyler [6], have used methods from quiver representation theory to show that, if W is a simply-laced, crystallographic Coxeter group which is irreducible and infinite then the word s1 · · · sns1 · · · sn · · · s1 · · · sn is reduced for any number of repetitions of s1 · · · sn. It is trivial to extend this result to the case where W ∼= W1 ×W2 × · · ·Wr, with each Wi a Coxeter group meeting the above conditions. The aim of this note is to reprove Kleiner and Pelley result using only the theory of Coxeter groups and the geometry of root systems. Our proof is inspired by that of Kleiner and Pelley, but we strip out the quiver theory and simplify several arguments. In the process, we strip out the assumptions that W is crystallographic and simply-laced. To repeat, our result is: Theorem 1. Let W be an infinite, irreducible Coxeter group and let (s1, · · · sn) be any ordering of the simple generators. Then the word s1 · · · sns1 · · · sn · · · s1 · · · sn is reduced for any number of repetitions of s1 · · · sn. More specifically, the authors of [8] define four properties of a sequence r1, r2, . . . of simple reflections; property (IV) is that the word r1r2 · · · rN is reduced for any N . For each classical affine type, they exhibit a sequence of reflections which satisfies their properties and the ordering is periodic in each case.