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BOUNDING REFLECTION LENGTH IN AN AFFINE COXETER GROUP
 

Summary: BOUNDING REFLECTION LENGTH IN
AN AFFINE COXETER GROUP
JON MCCAMMOND AND T. KYLE PETERSEN
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.
Every Coxeter group W has two natural generating sets: the set S
used in its standard presentation and the set R of reflections formed
by collecting all conjugates of the elements in S. The first generating
set leads to the standard length function S : W N and the second
is used to define the reflection length function R : W N. When

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics