Taking tilting modules from the poset of support tilting modules

TL;DR

This paper explores the relationship between tilting modules and support tilting modules within the framework of posets, demonstrating that the structure of tilting modules is uniquely determined by support tilting modules.

Contribution

It establishes that the subposet of tilting modules can be uniquely reconstructed from the poset of support tilting modules, linking combinatorial structures.

Findings

The subposet of tilting modules is uniquely determined by the poset of support tilting modules.

A combinatorial relationship between the two posets is established.

The study extends the partial order framework to support tilting modules.

Abstract

C. Ingalls and H. Thomas defined support tilting modules for path algebras. From tau-tilting theory introduced by T. Adachi, O. Iyama and I. Reiten, a partial order on the set of basic tilting modules defined by D. Happel and L. Unger is extended as a partial order on the set of support tilting modules. In this paper, we study a combinatorial relationship between the poset of basic tilting modules and basic support tilting modules. We will show that the subposet of tilting modules is uniquely determined by the poset structure of the set of support tilting modules.