Some Consequences of Categorification

Several conjectures on acyclic skew-symmetrizable cluster algebras are proven as direct consequences of their categorification via valued quivers. These include conjectures of Fomin-Zelevinsky, Reading-Speyer, and Reading-Stella related to $\mathbf{d}$-vectors, $\mathbf{g}$-vectors, and $F$-polynomials

Initial-seed recursions and dualities for d-vectors

We present an initial-seed-mutation formula for d-vectors of cluster variables in a cluster algebra. We also give two rephrasings of this recursion: one as a duality formula for d-vectors in the style of the g-vectors/c-vectors dualities of Nakanishi and Zelevinsky, and one as a formula expressing the highest powers in the Laurent expansion of a cluster variable in terms of the d-vectors of any cluster containing it. We prove that the initial-seed-mutation recursion holds in a varied collection of cluster algebras, but not in general. We conjecture further that the formula holds for source-sink moves on the initial seed in an arbitrary cluster algebra, and we prove this conjecture in the case of surfaces.Comment: 21 Pages, 20 Figures. Version 2: Expanded introduction, other minor expository changes. Version 3: Very minor corrections. Final version to appear in the Pacific Journal of Mathematic

Wonder of sine-Gordon Y-systems

The sine-Gordon Y-systems and the reduced sine-Gordon Y-systems were introduced by Tateo in the 90's in the study of the integrable deformation of conformal field theory by the thermodynamic Bethe ansatz method. The periodicity property and the dilogarithm identities concerning these Y-systems were conjectured by Tateo, and only a part of them have been proved so far. In this paper we formulate these Y-systems by the polygon realization of cluster algebras of types A and D, and prove the conjectured periodicity and dilogarithm identities in full generality. As it turns out, there is a wonderful interplay among continued fractions, triangulations of polygons, cluster algebras, and Y-systems.Comment: v1: 66 pages; v2: 53 pages, the version to appear in Trans. Amer. Math. Soc. (in the journal version, the proofs of Props. 5.29-5.31 and Sect. 5.8 will be omitted due to the limitation of space); v3: 53 pages, minor improvement of figures; v4 (no text changes): Sage (v7.0 and higher) has built-in functions to plot the triangulations associated with sine-Gordon and reduced sine-Gordon Y-system

An affine almost positive roots model

We generalize the almost positive roots model for cluster algebras from finite type to a uniform finite/affine type model. We define the almost positive Schur roots $\Phi_c$ and a compatibility degree, given by a formula that is new even in finite type. The clusters define a complete fan $\operatorname{Fan}_c(\Phi)$. Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of $\operatorname{Fan}_c(\Phi)$ induced by real roots to the ${\mathbf g}$-vector fan of the associated cluster algebra. We show that $\Phi_c$ is the set of denominator vectors of the associated acyclic cluster algebra and conjecture that the compatibility degree also describes denominator vectors for non-acyclic initial seeds. We extend results on exchangeability of roots to the affine case.Comment: 45 pages. *Version 4 addresses concerns from a referee * Version 3 corrects typesetting errors caused by the order of packages in the preamble * Version 2 is a major revision and contains only the results concerning the affine almost positive roots model; the discussion on orbits of coxeter elements is now arXiv:1808.0509

Polytopal realizations of finite type g\mathbf{g}-vector fans

This paper shows the polytopality of any finite type $\mathbf{g}$-vector fan, acyclic or not. In fact, for any finite Dynkin type $\Gamma$, we construct a universal associahedron $\mathsf{Asso}_{\mathrm{un}}(\Gamma)$ with the property that any $\mathbf{g}$-vector fan of type $\Gamma$ is the normal fan of a suitable projection of $\mathsf{Asso}_{\mathrm{un}}(\Gamma)$.Comment: 27 pages, 9 figures; Version 2: Minor changes in the introductio

The action of a Coxeter element on an affine root system

The characterization of orbits of roots under the action of a Coxeter element is a fundamental tool in the study of finite root systems and their reflection groups. This paper develops the analogous tool in the affine setting, adding detail and uniformity to a result of Dlab and Ringel.Comment: Version 1: This is a revised version of the first 1/3 of arXiv:1707.00340. (That paper will also be replaced by a new version, deleting the first 1/3 and with other major revisions). Version 2: Minor corrections to front and back matter. Version 3: Added mention of Dlab and Ringel, who proved essentially the same result in 1976 by a non-uniform argumen

The action of a Coxeter element on an affine root system

The characterization of orbits of roots under the action of a Coxeter element is a fundamental tool in the study of finite root systems and their reflection groups. This paper develops the analogous tool in the affine setting, adding detail and uniformity to a result of Dlab and Ringel

The greedy basis equals the theta basis

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 page