Tubes Containing String Modules in Symmetric Special Multiserial Algebras

TL;DR

This paper explores the module category of symmetric special multiserial algebras using combinatorial Brauer configurations, introducing Green hyperwalks to identify and analyze tubes in their Auslander-Reiten quivers.

Contribution

It introduces Green hyperwalks as a combinatorial tool to determine the existence, ranks, and structure of tubes in the stable Auslander-Reiten quiver of these algebras.

Findings

Green hyperwalks correspond to periodic projective resolutions.

Ranks of tubes are determined by periods of Green hyperwalks.

Explicit description of extensions in rank two tubes.

Abstract

Symmetric special multiserial algebras are algebras that correspond to decorated hypergraphs with orientation, called Brauer configurations. In this paper, we use the combinatorics of Brauer configurations to understand the module category of symmetric special multiserial algebras via their Auslander-Reiten quiver. In particular, we provide methods for determining the existence and ranks of tubes in the stable Auslander-Reiten quiver of symmetric special multiserial algebras using only the information from the underlying Brauer configuration. Firstly, we define a combinatorial walk around the Brauer configuration, called a Green `hyperwalk', which generalises the existing notion of a Green walk around a Brauer graph. Periodic Green hyperwalks are then shown to correspond to periodic projective resolutions of certain classes of string modules over the corresponding symmetric special multiserial algebra. Periodic Green hyperwalks thus determine certain classes of tubes in the stable Auslander-Reiten quiver, with the ranks of the tubes determined by the periods of the walks. Finally, we provide a description of additional rank two tubes in symmetric special multiserial algebras that do not arise from Green hyperwalks, but which nevertheless contain string modules at the mouth. This includes an explicit description of the space of extensions between string modules at the mouth of tubes of rank two.