On Commutation Classes of Reduced Words in Weyl Groups

This paper is concerned with commutation classes of reduced words in Weyl groups. It is divided in three sections. In the first, we give a recursive formula for the number of reduced words in a commutation class. In the second, we give a tableau-like description of all the reduced words adapted to a quiver in the case of simply laced root system. In the last, we consider the case of the longest element w0for the symmetric group S5and illustrate the fact that the set of commutation classes of reduced words of w0have nice symmetries and a topological structure.