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Symmetric special multiserial algebras are algebras that correspond todecorated hypergraphs with orientation, called Brauer configurations. In thispaper, we use the combinatorics of Brauer configurations to understand themodule category of symmetric special multiserial algebras via theirAuslander-Reiten quiver. In particular, we provide methods for determining theexistence and ranks of tubes in the stable Auslander-Reiten quiver of symmetricspecial multiserial algebras using only the information from the underlyingBrauer configuration. Firstly, we define a combinatorial walk around the Brauerconfiguration, called a Green `hyperwalk', which generalises the existingnotion of a Green walk around a Brauer graph. Periodic Green hyperwalks arethen shown to correspond to periodic projective resolutions of certain classesof string modules over the corresponding symmetric special multiserial algebra.Periodic Green hyperwalks thus determine certain classes of tubes in the stableAuslander-Reiten quiver, with the ranks of the tubes determined by the periodsof the walks. Finally, we provide a description of additional rank two tubes insymmetric special multiserial algebras that do not arise from Green hyperwalks,but which nevertheless contain string modules at the mouth. This includes anexplicit description of the space of extensions between string modules at themouth of tubes of rank two.

Tubes Containing String Modules in Symmetric Special Multiserial Algebras