Irreducible components of varieties of representations II

TL;DR

This paper classifies the irreducible components of representation varieties for a broad class of finite-dimensional algebras, providing generic descriptions of modules and their socle series, especially for truncated path algebras over acyclic quivers.

Contribution

It offers a characterization and listing method for irreducible components of representation varieties of truncated path algebras based on quiver and Loewy length, extending existing theory.

Findings

Classified irreducible components using quiver and Loewy length.

Described generic features of modules in each component.

Determined the generic socle series for modules in these components.

Abstract

The goals of this article are as follows: (1) To determine the irreducible components of the affine varieties parametrizing the representations of $ \Lambda $ with dimension vector d, where $ \Lambda $ traces a major class of finite dimensional algebras; (2) To generically describe the representations encoded by the components. The target class consists of those truncated path algebras $ \Lambda $ over an algebraically closed field K which are based on a quiver Q without oriented cycles. The main result characterizes the irreducible components of the representation variety in representation-theoretic terms and provides a means of listing them from quiver and Loewy length of $ \Lambda $. Combined with existing theory, this classification moreover yields an array of generic features of the modules parametrized by the components, such as generic minimal projective presentations, generic sub- and quotient modules, etc. Our second principal result pins down the generic socle series of the modules in the components; it does so for more general $ \Lambda $, in fact. The information on truncated path algebras of acyclic quivers supplements the theory available in the special case where $ \Lambda = KQ $, filling in generic data on the d-dimensional representations of Q with any fixed Loewy length.