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Salvatore Stella

Publications and source records attributed to Salvatore Stella.

26 records · Page 2Linked to original sources
arXiv2017

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.

math.CO↗
arXiv2017

On Generalized Minors and Quiver Representations

The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac-Moody group -- the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use this realization to connect representations of the quiver with those of the group. We show that cluster variables of preprojective (resp. postinjective) quiver representations are realized by generalized minors of highest-weight (resp. lowest-weight) group representations, generalizing results of Yang-Zelevinsky in finite type. In type $A_n^{\!(1)}$ and finitely many other affine types, we show that cluster variables of regular quiver representations are realized by generalized minors of group representations that are neither highest- nor lowest-weight; we conjecture this holds more generally.

math.RT↗
arXiv2016

SL_2-Tilings Do Not Exist in Higher Dimensions (mostly)

We define a family of generalizations of $\operatorname{SL}_2$-tilings to higher dimensions called $\boldsymbolε$-$\operatorname{SL}_2$-tilings. We show that, in each dimension 3 or greater, $\boldsymbolε$-$\operatorname{SL}_2$-tilings exist only for certain choices of $\boldsymbolε$. In the case that they exist, we show that they are essentially unique and have a concrete description in terms of odd Fibonacci numbers.

math.CO↗
arXiv2016

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.

math.QA↗
arXiv2015

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.

math.QA↗
arXiv2014

Diagrammatic description of c-vectors and d-vectors of cluster algebras of finite type

We provide an explicit Dynkin diagrammatic description of the c-vectors and the d-vectors (the denominator vectors) of any cluster algebra of finite type with principal coefficients and any initial exchange matrix. We use the surface realization of cluster algebras for types A_n and D_n, then we apply the folding method to D_{n+1} and A_{2n-1} to obtain types B_n and C_n. Exceptional types are done by direct inspection with the help of a computer algebra software. We also propose a conjecture on the root property of c-vectors for a general cluster algebra.

math.RA↗
arXiv2012

Polyhedral models for generalized associahedra via Coxeter elements

Motivated by the theory of cluster algebras, F. Chapoton, S. Fomin and A. Zelevinsky associated to each finite type root system a simple convex polytope called \emph{generalized associahedron}. They provided an explicit realization of this polytope associated with a bipartite orientation of the corresponding Dynkin diagram. In the first part of this paper, using the parametrization of cluster variables by their $g$-vectors explicitly computed by S.-W. Yang and A. Zelevinsky, we generalize the original construction to any orientation. In the second part we show that our construction agrees with the one given by C. Hohlweg, C. Lange, and H. Thomas in the setup of Cambrian fans developed by N. Reading and D. Speyer.

math.CO↗