Stability spaces of string and band modules

TL;DR

This paper explores the geometric structure of stability spaces for string and band modules, revealing their relationships and limits, and connecting them to order polytopes and thin modules.

Contribution

It introduces a dual perspective on stability spaces, relates them to order polytopes, and describes how band modules' stability spaces are limits of string modules' stability spaces.

Findings

Stability spaces of thin modules relate to order polytopes.

Stability spaces of band modules are unions of limits of string modules.

Band modules' stability spaces can be viewed as limits of string modules' stability spaces.

Abstract

The stability space of a module is the cone of vectors which make the module semistable. These cones are defined in terms of inequalities; in this paper we draw insights from considering the dual description in terms of non-negative linear spans. We show how stability spaces of thin modules are related to order polytopes. In the case of non-thin modules, we show how the stability spaces of string and band modules are related to the stability spaces of the thin modules corresponding to the abstract string and band. We use this to analyse the way in which the stability space of a band module is the limit of stability spaces of string modules. Namely, the stability space of the band module is a union of cones, each of which is the limit of the stability spaces of a family of string modules.