The geometry of uniserial representations of algebras II. Alternate viewpoints and uniqueness

TL;DR

This paper explores two geometric frameworks for modeling uniserial representations of finite-dimensional algebras, revealing invariance properties and fiber structures that enhance understanding of their geometric and algebraic features.

Contribution

It introduces two new geometric perspectives on uniserial varieties, including a Grassmannian-based model and a reinterpretation within module varieties, establishing invariance and fiber closedness.

Findings

Uniserial varieties form a quasi-projective subvariety of a Grassmannian.

Fibers of the canonical maps to uniserial representations are closed.

The new models provide intrinsic and invariant descriptions of uniserial representations.

Abstract

We provide two alternate settings for a family of varieties modeling the uniserial representations with fixed sequence of composition factors over a finite dimensional algebra. The first is a quasi-projective subvariety of a Grassmannian containing the members of the mentioned family as a principal affine open cover; among other benefits, one derives invariance from this intrinsic description. The second viewpoint re-interprets the `uniserial varieties' as locally closed subvarieties of the traditional module varieties; in particular, it exhibits closedness of the fibres of the canonical maps from the uniserial varieties to the uniserial representations.