The bijection between exceptional subcategories and non-crossing partitions

TL;DR

This paper presents a new proof of the bijection between exceptional subcategories of quiver representations and non-crossing partitions of Weyl groups, utilizing the exchange property of Weyl groups in Kac-Moody Lie algebras.

Contribution

It provides a novel proof of the bijection using the exchange property, enhancing understanding of the relationship between algebraic and combinatorial structures.

Findings

Established a new proof of the bijection

Connected exceptional subcategories with non-crossing partitions

Utilized properties of Weyl groups in the proof

Abstract

This note discusses the bijection between the exceptional subcategories of representations of quivers and generalized non-crossing partitions of Weyl groups. We give a new proof of the Ingalls-Thomas-Igusa-Schiffler bijection by using the exchange property of the Weyl groups of the Kac-Moody Lie algebras.