Ext-quivers of hearts of A-type and the orientation of associahedron

TL;DR

This paper classifies Ext-quivers of hearts in derived categories related to A-type quivers, linking combinatorial structures like binary trees to torsion pairs and cluster tilting sets, and relates associahedron orientations to these structures.

Contribution

It provides a classification of Ext-quivers for hearts in derived categories of A-type, connecting combinatorial and algebraic structures in cluster theory.

Findings

Classification of Ext-quivers for hearts in derived categories.

Explicit construction from binary trees to torsion pairs and cluster tilting sets.

Equivalence of associahedron orientations induced by different poset structures.

Abstract

We classify the Ext-quivers of hearts in the bounded derived category D(A_n) and the finite-dimensional derived category D(\Gamma_N A_n) of the Calabi-Yau-N Ginzburg algebra D(\Gamma_N A_n). This provides the classification for Buan-Thomas' colored quiver for higher clusters of A-type. We also give explicit combinatorial constructions from a binary tree with n+2 leaves to a torsion pair in mod k\overrightarrow{A_n} and a cluster tilting set in the corresponding cluster category, for the straight oriented A-type quiver \overrightarrow{A_n}. As an application, we show that the orientation of the n-dimensional ssociahedron induced by poset structure of binary trees coincides with the orientation induced by poset structure of torsion pairs in mod k\overrightarrow{A_n} (under the correspondence above).