Module varieties and representation type of finite-dimensional algebras

TL;DR

This paper characterizes finite-dimensional algebras with geometric properties like dense orbits and multiplicity-free conditions, linking these to classical notions of representation finiteness and providing new classifications.

Contribution

It introduces the dense-orbit and multiplicity-free properties for algebras and establishes their equivalence to representation-finiteness in various contexts, offering new geometric criteria.

Findings

String algebras with dense orbit-property are exactly the representation-finite ones.

An algebra can have the dense-orbit property without being representation-finite.

Tame algebras are Schur-representation-finite if and only if they have the multiplicity-free property.

Abstract

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.