  In this paper we seek geometric and invariant-theoretic characterizations of
(Schur-)representation finite algebras. To this end, we introduce two classes
of finite-dimensional algebras: those with the dense-orbit property and those
with the multiplicity-free property. We show first that when a connected
algebra A admits a preprojective component, each of these properties is
equivalent to A being representation-finite. Next, we give an example of an
algebra which is not representation-finite but still has the dense-orbit
property. We also show that the string algebras with the dense orbit-property
are precisely the representation-finite ones. Finally, we show that a tame
algebra has the multiplicity-free property if and only if it is
Schur-representation-finite.
