Indices and c-vectors in extriangulated categories

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.