  This paper studies connections between the preprojective representations of a
valued quiver, the (+)-admissible sequences of vertices, and the Weyl group by
associating to each preprojective representation a canonical (+)-admissible
sequence. A (+)-admissible sequence is the canonical sequence of some
preprojective representation if and only if the product of simple reflections
associated to the vertices of the sequence is a reduced word in the Weyl group.
As a consequence, for any Coxeter element of the Weyl group associated to an
indecomposable symmetrizable generalized Cartan matrix, the group is infinite
if and only if the powers of the element are reduced words. The latter
strengthens known results of Howlett, Fomin-Zelevinsky, and the authors.
