Cluster algebras of infinite rank as colimits

We formalize the way in which one can think about cluster algebras of infinite rank by showing that every rooted cluster algebra of infinite rank can be written as a colimit of rooted cluster algebras of finite rank. Relying on the proof of the posivity conjecture for skew-symmetric cluster algebras (of finite rank) by Lee and Schiffler, it follows as a direct consequence that the positivity conjecture holds for cluster algebras of infinite rank. Furthermore, we give a sufficient and necessary condition for a ring homomorphism between cluster algebras to give rise to a rooted cluster morphism without specializations. Assem, Dupont and Schiffler proposed the problem of a classification of ideal rooted cluster morphisms. We provide a partial solution by showing that every rooted cluster morphism without specializations is ideal, but in general rooted cluster morphisms are not ideal.Comment: Included cluster algebras of uncountable rank, fixed some typos. Results on the countable case unchanged, comments appreciate

Asymptotic sign coherence conjecture

The sign coherence phenomenon is an important feature of c-vectors in cluster algebras with principal coefficients. In this note, we consider a more general version of c-vectors defined for arbitrary cluster algebras of geometric type and formulate a conjecture describing their asymptotic behavior. This conjecture, which is called the asymptotic sign coherence conjecture, states that for any infinite sequence of matrix mutations that satisfies certain natural conditions, the corresponding c-vectors eventually become sign coherent. We prove this conjecture for rank 2 cluster algebras of infinite type and for a particular sequence of mutations in a cluster algebra associated with the Markov quiver.Comment: 13 pages, 2 figure

Graded quantum cluster algebras of infinite rank as colimits

We provide a graded and quantum version of the category of rooted cluster algebras introduced by Assem, Dupont and Schiffler and show that every graded quantum cluster algebra of infinite rank can be written as a colimit of graded quantum cluster algebras of finite rank. As an application, for each k we construct a graded quantum infinite Grassmannian admitting a cluster algebra structure, extending an earlier construction of the authors for k=2

Shard modules

Motivated by the goal of studying cluster algebras in infinite type, we study the stability domains of modules for the preprojective algebra in the corresponding infinite types. Specifically, we study real bricks: those modules whose endomorphism algebra is a division ring and which have no self-extensions. We define "shard modules" to be those real bricks whose stability domain is as large as possible (meaning, of dimension one less than the rank of the preprojective algebra). We show that all real bricks are obtained by applying the Baumann-Kamnitzer reflection functors to simple modules, and we give a recursive formula for the stability domain of a real brick. We show that shard modules are in bijection with Nathan Reading's "shards", and that their stability domains are the shards; we also establish many foundational results about shards in infinite type which have not previously appeared in print. With an eye toward applications to cluster algebras, our paper is written to handle skew-symmetrizable as well as skew-symmetric exchange matrices, and we therefore discuss the basics of the theory of species for preprojective algebras. We also give some counterexamples to show ways in which infinite type is more subtle than the well-studied finite type cases.Comment: 40 page