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Classification of cluster variables in cluster algebras (in particular, Grassmannian cluster algebras) is an important problem, which has direct application to computations of scattering amplitudes in physics. In this paper, we apply the tableaux method to classify cluster variables in Grassmannian cluster algebras $\mathbb{C}[Gr(k,n)]$ up to $(k,n)=(3,12), (4,10)$, or $(4,12)$ up to a certain number of columns of tableaux, using HPC clusters. These datasets are made available on GitHub. Supervised and unsupervised machine learning methods are used to analyse this data and identify structures associated to tableaux corresponding to cluster variables. Conjectures are raised associated to the enumeration of tableaux at each rank and the tableaux structure which creates a cluster variable, with the aid of machine learning., Comment: 32 pages; 14 figures

We construct special Lagrangian fibrations for log Calabi-Yau surfaces, and scattering diagrams from Lagrangian Floer theory of the fibres. Then we prove that the scattering diagrams recover the scattering diagrams of Gross-Pandharipande-Siebert and the canonical scattering diagrams of Gross-Hacking-Keel. With an additional assumption on the non-negativity of boundary divisors, we compute the disc potentials of the Lagrangian torus fibres via a holomorphic/tropical correspondence. As an application, we provide a version of mirror symmetry for rank two cluster varieties., Comment: 61 pages, 9 figures. comments are welcome; v2. exposition improved, accepted version

We give a construction of generalized cluster varieties and generalized cluster scattering diagrams for reciprocal generalized cluster algebras, the latter of which were defined by Chekhov and Shapiro. These constructions are analogous to the structures given for ordinary cluster algebras in the work of Gross, Hacking, Keel, and Kontsevich. As a consequence of these constructions, we are also able to construct theta functions for generalized cluster algebras, again in the reciprocal case, and demonstrate a number of their structural properties., Comment: Updated to reflect reviewers' comments and to use the preferred journal formatting

Fock and Goncharov described a quantization of cluster $\mathcal{X}$-varieties (also known as cluster Poisson varieties) in [FG09]. Meanwhile, families of deformations of cluster $\mathcal{X}$-varieties were introduced in [BFMNC18]. In this paper we show that the two constructions are compatible -- we extend the Fock-Goncharov quantization of $\mathcal{X}$-varieties to the families of [BFMNC18]. As a corollary, we obtain that these families and each of their fibers have Poisson structures. We relate this construction to the Berenstein-Zelevinsky quantization of $\mathcal{A}$-varieties ([BZ05]). Finally, inspired by the counter-example to quantum positivity of the quantum greedy basis in [LLRZ14], we compute a counter-example to quantum positivity of the quantum theta basis., Comment: 46 pages, 4 figures, 2 tables. We added Definition 3.5 in order to obtain a correct proof of Proposition 3.6 (formerly Proposition 3.5). We added a discussion of Definition 3.11 (formerly Definition 3.10) in which we describe the gluings in greater detail. Other minor corrections. To appear in Algebras and Representation Theory

In [GHKK18], Gross-Hacking-Keel-Kontsevich discuss compactifications of cluster varieties from "positive subsets" in the real tropicalization of the mirror. To be more precise, let $\mathfrak{D}$ be the scattering diagram of a cluster variety $V$ (of either type -- $\mathcal{A}$ or $\mathcal{X}$), and let $S$ be a closed subset of $\left(V^\vee\right)^{\text{trop}}(\mathbb{R})$ -- the ambient space of $\mathfrak{D}$. The set $S$ is positive if the theta functions corresponding to the integral points of $S$ and its $\mathbb{N}$-dilations define an $\mathbb{N}$-graded subalgebra of $\Gamma(V, \mathcal{O}_V)[x]$. In particular, a positive set $S$ defines a compactification of $V$ through a Proj construction applied to the corresponding $\mathbb{N}$-graded algebra. In this paper we give a natural convexity notion for subsets of $\mathfrak{D}$, called "broken line convexity", and show that a set is positive if and only if it is broken line convex. The combinatorial criterion of broken line convexity provides a tractable way to construct positive subsets of $\mathfrak{D}$, or to check positivity of a given subset., Comment: 40 pages, 19 figures, comments welcome. Added subsection 2.2.3 treating quotients and fibers of cluster varieties. The main theorem holds in this broader setting. To appear in IMRN

In this article, we use the relationship between cluster scattering diagrams and stability scattering diagrams to relate quiver representations with these diagrams. With a notion of positive crossing of a path $\gamma$, we show that if $\gamma$ has positive crossing in the scattering diagram, then it goes in the opposite direction of the Auslander-Reiten quiver of $Q$. We then give the Hall algebra theta functions which recover the cluster character formula by the Euler characteristic map. At last, we define the Hall algebra broken lines and then are able to give the stratification of the quiver Grassmannians by the bending of the broken lines., Comment: 40 pages, 5 figures

We prove the equality of two canonical bases of a rank 2 cluster algebra, the greedy basis of Lee-Li-Zelevinsky and the theta basis of Gross-Hacking-Keel-Kontsevich., Comment: 17 pages