BGP-reflection functors and cluster combinatorics

TL;DR

This paper introduces Bernstein-Gelfand-Ponomarev reflection functors within cluster categories, providing a quiver-based realization of root system symmetries and confirming related conjectures across all Dynkin types.

Contribution

It defines new reflection functors in cluster categories, linking them to root system symmetries and confirming a key conjecture for all Dynkin types.

Findings

Provides a quiver realization of simple reflections on roots

Confirms conjecture 9.1 in all Dynkin types

Unifies interpretation of generalized associahedra

Abstract

We define Bernstein-Gelfand-Ponomarev reflection functors in the cluster categories of hereditary algebras. They are triangle equivalences which provide a natural quiver realization of the "truncated simple reflections" on the set of almost positive roots $\Phi_{\ge -1}$ associated to a finite dimensional semisimple Lie algebra. Combining with the tilting theory in cluster categories developed in [4], we give a unified interpretation via quiver representations for the generalized associahedra associated to the root systems of all Dynkin types (a simply-laced or non-simply-laced). This confirms the conjecture 9.1 in [4] in all Dynkin types.