A Characterization of One-Element Commutation Classes

Abstract

A reduced word for a permutation of the symmetric group is its own commutation class if it has no commutation moves available. These words have the property that every factor of length 2 is formed by consecutive integers, but in general words of this form may not be reduced. In this paper we give a necessary and sufficient condition for a word with the previous property to be reduced. In the case of involutions, we give an explicitly construction of their one-element commutation classes and relate their existence with pattern avoidance problems.