  This paper concerns preprojective representations of a finite connected
valued quiver without oriented cycles. For each such representation, an
explicit formula in terms of the geometry of the quiver gives a unique, up to a
certain equivalence, shortest (+)-admissible sequence such that the
corresponding composition of reflection functors annihilates the
representation. The set of equivalence classes of the above sequences is a
partially ordered set that contains a great deal of information about the
preprojective component of the Auslander-Reiten quiver. The results apply to
the study of reduced words in the Weyl group associated to an indecomposable
symmetrizable generalized Cartan matrix.
