Starting from an arbitrary cluster-tilting object T in a 2-Calabi-Yau category over an algebraically closed field, as in the setting of Keller and Reiten, we define, for each object L, a fraction X(T, L) using a formula proposed by Caldero and Keller. We show that the map taking L to X(T, L) is a cluster character, i.e. that it satisfies a certain multiplication formula. We deduce that it induces a bijection, in the finite and the acyclic case, between the indecomposable rigid objects of the cluster category and the cluster variables, which confirms a conjecture of Caldero and Keller.
We show that the mesh mutations are the minimal relations among the g-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 g-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 N. Arkani-Hamed, Y. Bai, S. He, and G. Yan in type A and of V. Bazier-Matte, G. Douville, K. Mousavand, H. Thomas and E. Yıldırım for acyclic initial seeds.Moreover, we use a similar approach to study the space of polytopal realizations of the g-vector fans of other generalizations of the associahedron. For non-kissing complexes (a.k.a. support τ -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. For graphical nested complexes, we show that the space of realizations of the nested fan is simplicial only in the case of the associahedron, and we describe its facet-defining inequalities in general.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 M. Auslander on minimal relations for Grothendieck groups of module categories.The type cone of the cluster fan of type A 2 .
We give a simultaneous generalization of exact categories and triangulated categories, which is suitable for considering cotorsion pairs, and which we call extriangulated categories. Extension-closed, full subcategories of triangulated categories are examples of extriangulated categories. We give a bijective correspondence between some pairs of cotorsion pairs which we call Hovey twin cotorsion pairs, and admissible model structures. As a consequence, these model structures relate certain localizations with certain ideal quotients, via the homotopy category which can be given a triangulated structure. This gives a natural framework to formulate reduction and mutation of cotorsion pairs, applicable to both exact categories and triangulated categories. These results can be thought of as arguments towards the view that extriangulated categories are a convenient setup for writing down proofs which apply to both exact categories and (extension-closed subcategories of) triangulated categories.
We interpret the support τ \tau -tilting complex of any gentle bound quiver as the non-kissing complex of walks on its blossoming quiver. Particularly relevant examples were previously studied for quivers defined by a subset of the grid or by a dissection of a polygon. We then focus on the case when the non-kissing complex is finite. We show that the graph of increasing flips on its facets is the Hasse diagram of a congruence-uniform lattice. Finally, we study its g \mathbf {g} -vector fan and prove that it is the normal fan of a non-kissing associahedron.
We interpret the support τ -tilting complex of any gentle bound quiver as the nonkissing complex of walks on its blossoming quiver. Particularly relevant examples were previously studied for quivers defined by a subset of the grid or by a dissection of a polygon. We then focus on the case when the non-kissing complex is finite. We show that the graph of increasing flips on its facets is the Hasse diagram of a congruence-uniform lattice. Finally, we study its g-vector fan and prove that it is the normal fan of a non-kissing associahedron. ContentsPart 2. The non-kissing complex 2.1. Blossoming quivers 2.1.1. Blossoming quiver of a gentle bound quiver 2.1.2. Dissection and grid bound quivers 2.2. The non-kissing complex 2.2.1. The non-kissing complex 2.2.2. Distinguished walks, arrows and substrings 2.2.3. Flips 2.3. Non-kissing complexes versus support τ -tilting complexes 2.3.1. From walks to strings 2.3.2. From walks to AR-quivers The three authors were partially supported by the French ANR grant SC3A (15 CE40 0004 01).
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