Arc diagrams and 2-term simple-minded collections of preprojective algebras of type $A$

TL;DR

This paper characterizes semibricks and 2-term simple-minded collections of preprojective algebras of type A using arc diagrams, establishing bijections with symmetric group elements and exploring their mutation and order structures.

Contribution

It provides explicit combinatorial descriptions and bijections for semibricks and simple-minded collections via arc diagrams, linking algebraic structures to symmetric group combinatorics.

Findings

Bijection between noncrossoing arc diagrams and semibricks

Poset isomorphism between symmetric group and 2-term simple-minded collections

Application to quotient algebras and reproof of existing results

Abstract

We study an explicit description of semibricks and 2-term simple-minded collections over preprojective algebras of type $A$ via arc diagrams. We provide a bijection between the set of noncrossoing arc diagrams (resp. the set of double arc diagrams), which is in bijective correspondence with elements of the symmetric group, and the set of semibricks (resp. the set of 2-term simple-minded collections) over the algebra. Moreover we define a mutation and a partial order on the set of double arc diagrams. In particular, we obtain a poset isomorphism between the symmetric group and the set of 2-term simple-minded collections. As an application of our results, we study semibricks of some quotient algebras of the preprojective algebras of type $A$ and we reprove some important results shown by the other authors.