On algebras of finite general representation type

TL;DR

This paper introduces the concept of finite general representation type for finite-dimensional algebras, exploring its properties, examples, and conjectures, and distinguishes it from traditional representation types using geometric and algebraic methods.

Contribution

It defines finite general representation type, provides examples of wild algebras with this property, and proposes a related Brauer-Thrall style conjecture.

Findings

Identified algebras of wild type with finite general representation type

Proved local algebras of discrete general representation type are of finite type

Formulated a conjecture for general representations of algebras

Abstract

We introduce the notion of ``finite general representation type'' for a finite-dimensional algebra, a property related to the ``dense orbit property'' introduced by Chindris-Kinser-Weyman. We use an interplay of geometric, combinatorial, and algebraic methods to produce a family of algebras of wild representation type but finite general representation type. For completeness, we also give a short proof that the only local algebras of discrete general representation type are already of finite representation type. We end with a Brauer-Thrall style conjecture for general representations of algebras.