On simple-minded systems over representation-finite self-injective algebras

TL;DR

This paper characterizes simple-minded systems in the stable module category of representation-finite self-injective algebras and shows that Nakayama-stable orthogonal systems can be extended to such systems.

Contribution

It provides a new characterization of simple-minded systems and proves that Nakayama-stable orthogonal systems always extend to them.

Findings

New characterization for simple-minded systems in $A$-$ ext{stmod}$

Every Nakayama-stable orthogonal system extends to a simple-minded system

Advances understanding of the structure of stable module categories

Abstract

Let $A$ be a representation-finite self-injective algebra over an algebraically closed field $k$. We give a new characterization for an orthogonal system in the stable module category $A$-$\stmod$ to be a simple-minded system. As a by-product, we show that every Nakayama-stable orthogonal system in $A$-$\stmod$ extends to a simple-minded system.