Reduced expression of minimal infinite reduced words of affine Weyl groups

Abstract

For an infinite Coxeter system, one can extend the weak right order to the set of equivalence classes of infinite reduced words. This is called limit weak order. In Lam and Pylyavskyy (2013) [12], Lam and Pylyavskyy showed that for affine Weyl groups of type A˜n, minimal elements of the limit weak order are equivalent to infinite Coxeter elements and asked the question of characterization, in terms of infinite reduced words, of the minimal elements of the limit weak order for other affine types. In this paper we give such a characterization by determining the reduced expression of a representative infinite reduced word for each of these minimal elements.