  It is known that the set of irreducible components of nilpotent varieties
provides a geometric realization of the crystal basis for quantum groups. For
each reduced expression of a Weyl group element, Gei{\ss}, Leclerc and
Schr\"{o}er has recently given a parametrization of irreducible components of
nilpotent varieties in studying cluster algebras. In this paper we show that
their parametrization coincides with Lusztig's parametrization of the canonical
basis.
