Birge Huisgen-Zimmermann

Classifying representations with squarefree top
by way of Grassmannians

Abstract.
Let $\Lambda$ be a finite dimensional algebra. It is proved
that existence of a coarse moduli space classifying, up to
isomorphism, the representations of $\Lambda$ with fixed 
dimension $d$ and fixed squarefree top $T$ is equivalent to
the existence of a fine moduli space for this classification
problem. Accessible criteria are given to characterize this
event, and in the positive case -- unexpectedly frequent in
light of the stringency of fine classification -- the
(necessarily projective) classifying variety is explicitly
described as a subvariety of a Grassmannian, which can be
computed from quiver and relations of $\Lambda$. Even when
the moduli problem fails to be solvable, this latter variety
is seen to have distinctive properties recommending it as a
substitute for a moduli space. As an application, a 
characterization of the algebras having only finitely many
representation with a fixed simple top is obtained.