Indices and c-vectors in extriangulated categories

Let C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal C}$$\end{document} be an extriangulated category and τ be any n-cluster tilting subcategory of C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal C}$$\end{document}. We consider the index with respect to τ and introduce the index Grothendieck group of τ. Using the index, we prove that the index Grothendieck group of τ is isomorphic to the Grothendieck group of C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal C}$$\end{document}, 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.