Complete \(g\)-fans of rank 2

\(g\)-fan of a finite dimensional algebra is a fan in its real Grothendieck group defined by tilting theory. We give a classification of complete \(g\)-fans of rank 2. More explicitly, our main result asserts that every complete sign-coherent fan of rank 2 is a \(g\)-fan of some finite dimensional algebra. Our proof is based on three fundamental results, Gluing Theorem, Rotation Theorem and Subdivision Theorem, which realize basic operations on fans in the level of finite dimensional algebras. This is a joint work with T. Aoki, A. Higashitani, O. Iyama and R. Kase.