Associahedra for finite-type cluster algebras and minimal relations between g-vectors

We show that the mesh mutations are the minimal relations among the (Formula presented.) -vectors with respect to any initial seed in any finite-type cluster algebra. We then use this algebraic result to derive geometric properties of the (Formula presented.) -vector fan: we show that the space of all its polytopal realizations is a simplicial cone, and we then observe that this property implies that all its realizations can be described as the intersection of a high-dimensional positive orthant with well-chosen affine spaces. This sheds a new light on and extends earlier results of Arkani-Hamed, Bai, He, and Yan in type (Formula presented.) and of Bazier-Matte, Chapelier-Laget, Douville, Mousavand, Thomas, and Yıldırım for acyclic initial seeds. Moreover, we use a similar approach to study the space of polytopal realizations of the (Formula presented.) -vector fans of another generalization of the associahedron: nonkissing complexes (also known as support (Formula presented.) -tilting complexes) of gentle algebras. We show that the space of realizations of the nonkissing fan is simplicial when the gentle bound quiver is brick and 2-acyclic, and we describe in this case its facet-defining inequalities in terms of mesh mutations. Along the way, we prove algebraic results on 2-Calabi–Yau triangulated categories, and on extriangulated categories that are of independent interest. In particular, we prove, in those two setups, an analogue of a result of Auslander on minimal relations for Grothendieck groups of module categories.