From support $\tau$-tilting posets to algebras

TL;DR

This paper investigates the structure of support τ-tilting posets in relation to certain classes of algebras, providing conditions for when different algebras share the same support τ-tilting poset.

Contribution

It establishes a correspondence between support τ-tilting posets and algebraic properties, offering necessary and sufficient conditions for poset equivalence in specific algebra classes.

Findings

Identifies algebraic conditions for support τ-tilting poset isomorphism.

Provides a characterization for algebras sharing the same support τ-tilting poset.

Applies results to preprojective, Nakayama, and Brauer tree algebras.

Abstract

The aim of this paper is to study a poset isomorphism between two support $\tau$-tilting posets. We take several algebraic information from combinatorial properties of support $\tau$-tilting posets. As an application, we treat a certain class of basic algebras which contains preprojective algebras of type $A$, Nakayama algebras, and generalized Brauer tree algebras. We provide a necessary condition for that an algebra $\Lambda$ share the same support $\tau$-tilting poset with a given algebra $\Gamma$ in this class. Furthermore, we see that this necessary condition is also a sufficient condition if $\Gamma$ is either a preprojective algebra of type $A$, a Nakayama algebra, or a generalized Brauer tree algebra.