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

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

2012-02-28

arxiv.org/abs/1202.6325v2

Mathematics

Representation Theory

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Scientific paper

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).

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