Ext-quivers of hearts of A-type and the orientations of associahedrons

Abstract

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