Two-term tilting complexes and simple-minded systems of self-injective Nakayama algebras

TL;DR

This paper explores the connection between simple-minded systems and two-term tilting complexes in self-injective Nakayama algebras, revealing that simple-minded systems can be obtained via stable equivalences induced by tilting complexes.

Contribution

It demonstrates that all simple-minded systems in these algebras are images of simple modules under stable equivalences from two-term tilting complexes.

Findings

Every simple-minded system arises from a two-term tilting complex.

Connections established between mutation theories, Brauer trees, and polygon triangulations.

Provides a unified framework linking combinatorics and algebraic structures.

Abstract

We study the relation between simple-minded systems and two-term tilting complexes for self-injective Nakayama algebras. More precisely, we show that any simple-minded system of a self-injective Nakayama algebra is the image of the set of simple modules under a stable equivalence, which is given by the restriction of a standard derived equivalence induced by a two-term tilting complex. We achieve this by exploiting and connecting the mutation theories from the combinatorics of Brauer tree, configurations of stable translations quivers of type A, and triangulations of a punctured convex regular polygon.