The cones of g-vectors

Abstract

This paper studies the wall-chamber structures of finite-dimensional ($τ$-tilting infinite) algebras via generic decompositions of g-vectors. In particular, we examine regions outside the chambers. We show that the cones of g-vectors are rational and simplicial. Moreover, we prove that the open cone of a given g-vector coincides with the interior of its $\TF$-equivalence class if and only if the two have the same dimension. Furthermore, we establish that g-vectors satisfy the ray condition when they are sufficiently far from the origin. As an application, we generalize several results of Asai and Iyama concerning $\TF$-equivalence classes of g-vectors.