Bounding reflection length in an affine Coxeter group

Details Bounding reflection length in an affine Coxeter group Bounding reflection length in an affine Coxeter group

2010-09-24

arxiv.org/abs/1009.4918v2

Mathematics

Combinatorics

10 pages. Replaces earlier posting by second author. Paper is substantially reorganized and includes stronger results, includi

Scientific paper

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 $\R^n$ 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.

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