Variations of the bridge quiver for domestic string algebras

TL;DR

This paper explores variations of the bridge quiver in domestic string algebras, introducing new structures like weak arch bridges and operations to facilitate the computation of representation-theoretic invariants.

Contribution

It extends the combinatorial framework of the bridge quiver by defining weak arch bridges and a new operation, forming a finite category for better analysis.

Findings

Defined weak arch bridges and their partial operation $ extcircled{H}$

Established a finite category structure with unique factorization into arch bridges

Facilitated computation of representation-theoretic invariants

Abstract

In the computation of some representation-theoretic numerical invariants of domestic string algebras, a finite combinatorial gadget introduced by Schr\"{o}er--the \emph{bridge quiver} whose vertices are (representatives of cyclic permutations of) bands and whose arrows are certain band-free strings--has been used extensively. There is a natural but ill-behaved partial binary operation, $\circ$, on the larger set of \emph{weak bridges} such that bridges are precisely the $\circ$-irreducibles. With the goal of computing hammocks up to isomorphism in a later work we equip an even larger set of \emph{weak arch bridges} with another partial binary operation, $\circ_H$, to obtain a finite category. Each weak arch bridge admits a unique $\circ_H$-factorization into \emph{arch bridges}, i.e., the $\circ_H$-irreducibles.