Let C be an extriangulated category and T be any n-cluster tilting subcategory of C.We consider the index with respect to T and introduce the index Grothendieck group of T.Using the index,we prove that the index Groth...Let C be an extriangulated category and T be any n-cluster tilting subcategory of C.We consider the index with respect to T and introduce the index Grothendieck group of T.Using the index,we prove that the index Grothendieck group of T is isomorphic to the Grothendieck group of C,which implies that the index Grothendieck groups of any two n-cluster tilting subcategories are isomorphic.In particular,we show that the split Grothendieck groups of any two 2-cluster tilting subcategories are isomorphic.Then we develop a general framework for c-vectors of 2-Calabi-Yau extriangulated categories with respect to arbitrary 2-cluster tilting subcategories.We show that the c-vectors have the sign-coherence property and provide some formulas for calculating c-vectors.展开更多
We introduce diagrams for m-cluster categories which we call "horizontal" and "vertical" mutation fans. These are analogous to the mutation fans(also known as "semi-invariant pictures" or...We introduce diagrams for m-cluster categories which we call "horizontal" and "vertical" mutation fans. These are analogous to the mutation fans(also known as "semi-invariant pictures" or "scattering diagrams") for the standard(m = 1) cluster case, which are dual to the poset of finitely generated torsion classes. The purpose of these diagrams is to visualize mutations and analogues of maximal green sequences in the m-cluster category with special emphasis on the c-vectors(the "brick" labels).展开更多
基金supported by National Natural Science Foundation of China(Grant No.12271257)。
文摘Let C be an extriangulated category and T be any n-cluster tilting subcategory of C.We consider the index with respect to T and introduce the index Grothendieck group of T.Using the index,we prove that the index Grothendieck group of T is isomorphic to the Grothendieck group of C,which implies that the index Grothendieck groups of any two n-cluster tilting subcategories are isomorphic.In particular,we show that the split Grothendieck groups of any two 2-cluster tilting subcategories are isomorphic.Then we develop a general framework for c-vectors of 2-Calabi-Yau extriangulated categories with respect to arbitrary 2-cluster tilting subcategories.We show that the c-vectors have the sign-coherence property and provide some formulas for calculating c-vectors.
文摘We introduce diagrams for m-cluster categories which we call "horizontal" and "vertical" mutation fans. These are analogous to the mutation fans(also known as "semi-invariant pictures" or "scattering diagrams") for the standard(m = 1) cluster case, which are dual to the poset of finitely generated torsion classes. The purpose of these diagrams is to visualize mutations and analogues of maximal green sequences in the m-cluster category with special emphasis on the c-vectors(the "brick" labels).
