Module varieties and representation type of finite-dimensional algebras

Details Module varieties and representation type of finite-dimensional algebras Module varieties and representation type of finite-dimensional algebras

2012-01-31

arxiv.org/abs/1201.6422v2

Mathematics

Representation Theory

24 pp., comments welcome. v2: simplified Section 3

Scientific paper

In this paper we seek 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 pre-projective component, each of these properties is equivalent to $A$ being representation-finite. Next, we give an example of a representation-infinite algebra with 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.

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