Understanding finite dimensional representations generically

TL;DR

This paper surveys the development of generic representation theory for finite dimensional algebras, focusing on classifying modules and understanding their properties across different algebra types.

Contribution

It provides a comprehensive overview of the progress and current state of generic representation theory for finite dimensional algebras since the 1980s.

Findings

Classification of irreducible components of module varieties

Description of generic module properties within components

Progress across hereditary, tame non-hereditary, and wild non-hereditary algebras

Abstract

We survey the development and status quo of a subject best described as "generic representation theory of finite dimensional algebras", which started taking shape in the early 1980s. Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field. Roughly, the theory aims at (a) pinning down the irreducible components of the standard parametrizing varieties for the $\Lambda$-modules with a fixed dimension vector, and (b) assembling generic information on the modules in each individual component, that is, assembling data shared by all modules in a dense open subset of that component. We present an overview of results spanning the spectrum from hereditary algebras through the tame non-hereditary case to wild non-hereditary algebras.