Reduction Techniques to Identify Connected Components of Mutation Quivers

TL;DR

This paper investigates the structure of mutation quivers in $ au$-tilting theory, providing methods to identify connected components and demonstrating classes of algebras with limited or multiple components.

Contribution

It introduces reduction techniques to determine connected components in mutation quivers of support $ au$-tilting pairs, extending previous results and providing new examples.

Findings

Algebras with two simple modules have at most two connected components in their mutation quivers.

The paper generalizes previous results by Demonet, Iyama, and Jasso (2017).

Examples of algebras with more than two components are constructed.

Abstract

Important objects of study in $\tau$-tilting theory include the $\tau$-tilting pairs over an algebra on the form $kQ/I$, with $kQ$ being a path algebra and $I$ an admissible ideal. In this paper, we study aspects of the combinatorics of mutation quivers of support $\tau$-tilting pairs, simply called mutation quivers. In particular, we are interested in identifying connected components of the underlying graphs of such quivers. We give a class of algebras with two simple modules such that every algebra in the class has at most two connected components in its mutation quiver, generalizing a result by Demonet, Iyama and Jasso (2017). We also give examples of algebras with strictly more than two components in their mutation quivers.