Bounding reflection length in an affine Coxeter group

2010/09/24 by McCammond, Jon, Petersen, T. Kyle
#20F55 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1009.4918

Abstract

In any Coxeter group, the conjugates of elements in the standard minimal generating set are called reflections and the minimal number of reflections needed to factor a particular element is called its reflection length. In this article we prove that the reflection length function on an affine Coxeter group has a uniform upper bound. More precisely we prove that the reflection length function on an affine Coxeter group that naturally acts faithfully and cocompactly on \Rⁿ is bounded above by 2n and we also show that this bound is optimal. Conjecturally, spherical and affine Coxeter groups are the only Coxeter groups with a uniform bound on reflection length.

Related

• Stable reflection length in Coxeter groups
• Powers of Coxeter elements with unbounded reflection length
• Reflection length at infinity in hyperbolic reflection groups
• A refined count of Coxeter element factorizations
• The depth of a permutation