A combinatorial procedure for tilting mutation

TL;DR

This paper introduces a combinatorial method for tilting mutation in algebra, enabling explicit transformations between certain derived equivalent algebras, exemplified by line and rectangle shaped quivers.

Contribution

It provides a new combinatorial procedure for tilting mutation applicable to suitable algebras, facilitating explicit derived equivalences.

Findings

Recreated Ladkani's derived equivalence result

Developed a combinatorial tilting mutation procedure

Explicitly connected line and rectangle quiver algebras

Abstract

Tilting mutation is a way of producing new tilting complexes from old ones replacing only one indecomposable summand. In this paper, we give a purely combinatorial procedure for performing tilting mutation of suitable algebras. As an application, we recreate a result due to Ladkani, which states that the path algebra of a quiver shaped like a line (with certain relations) is derived equivalent to the path algebra of a quiver shaped like a rectangle. We will do this by producing an explicit series of tilting mutations going between the two algebras.