Reduced words in affine Coxeter groups

Abstract

Let r( w) denote the number of reduced words for an element w in a Coxeter group W. Stanley proved a formula for r( w) when W is the symmetric group A n , and he suggested looking at r( w) for the affine group A n . We prove that for any affine Coxeter group X n there is a finite number of types of elements in X n , such that to every element w can be associated (1) a type t, (2) an element v in the finite group X n , and (3) an n-tuple ( m 1, m 2,…, m n ) of integers m i ⩾ 0. Then r( w) = r v t ( m 1,…, m n ), and for every r v t and for large enough m i , a homogeneous linear n-dimensional recurrence holds. For A n , this takes a nice combinatorial form. We also discuss a canonical reduced word for w associated to its n-tuple.