Salvatore Stella
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Salvatore Stella has a total of 22 co-authors. Here are the top co-authors by number of shared publications:
- Nathan Reading
Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205
5 shared publications - Dylan Rupel
Department of Mathematics, University of Notre Dame, Notre Dame, USA
4 shared publications Dylan Rupel
3 shared publications- Tomoki Nakanishi
Graduate School of Mathematics, Nagoya University
3 shared publications - Vincent Pilaud
Universitat de Barcelona & Centre de Recerca Matemàtica, Barcelona, Spain
3 shared publications - Christophe Hohlweg
Lacim, Université du Québec à Montréal, Montréal, Quebec, Canada
2 shared publications
Latest Publications
- 2 November 2026
- Published by MathDoc/Centre Mersenne in Comptes Rendus Mathématique
- Vol. 364 (G3) , 701-742
Cluster scattering diagrams of acyclic affine type
We give an explicit construction of the cluster scattering diagram for any acyclic exchange matrix of affine type. We show that the corresponding cluster scattering fan coincides both with the mutation fan and with a fan constructed in the almost positive roots model.
- 2 November 2026
- Published by MathDoc/Centre Mersenne in Comptes Rendus Mathématique
- Vol. 364 (G3) , 701-742
Cluster scattering diagrams of acyclic affine type
We give an explicit construction of the cluster scattering diagram for any acyclic exchange matrix of affine type. We show that the corresponding cluster scattering fan coincides both with the mutation fan and with a fan constructed in the almost positive roots model.
- 16 April 2020
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 148 (7) , 2783-2798
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.
- 16 April 2020
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 148 (7) , 2783-2798
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.
- 28 February 2020
- Published by European Mathematical Society - EMS - Publishing House GmbH in Journal of Combinatorial Algebra
- Vol. 4 (1) , 1-59
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 a subset \Phi_c of the root system and a compatibility degree on \Phi_c , given by a formula that is new even in finite type. The clusters (maximal pairwise compatible sets of roots) define a complete fan Fan _c(\Phi) . Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of 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.
- 28 February 2020
- Published by European Mathematical Society - EMS - Publishing House GmbH in Journal of Combinatorial Algebra
- Vol. 4 (1) , 1-59
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 a subset \Phi_c of the root system and a compatibility degree on \Phi_c , given by a formula that is new even in finite type. The clusters (maximal pairwise compatible sets of roots) define a complete fan Fan _c(\Phi) . Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of 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.
- Open Access
- 30 January 2020
- Published by SIGMA (Symmetry, Integrability and Geometry: Methods and Application) in Symmetry, Integrability and Geometry: Methods and Applications
- Vol. 16
Some Consequences of Categorification
- 30 January 2020
- Published by SIGMA (Symmetry, Integrability and Geometry: Methods and Application) in Symmetry, Integrability and Geometry: Methods and Applications
- Vol. 16
Open AccessSome Consequences of Categorification
- 14 June 2019
- Published by Cambridge University Press (CUP) in Compositio Mathematica
- Vol. 155 (7) , 1301-1326
Affine cluster monomials are generalized minors
We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac–Moody groups. We prove that all cluster monomials with $\mathbf{g}$ -vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero–Chapoton description via quiver representations. In type $A_{1}^{(1)}$ , we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac–Moody algebras.- 14 June 2019
- Published by Cambridge University Press (CUP) in Compositio Mathematica
- Vol. 155 (7) , 1301-1326
Affine cluster monomials are generalized minors
We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac–Moody groups. We prove that all cluster monomials with $\mathbf{g}$ -vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero–Chapoton description via quiver representations. In type $A_{1}^{(1)}$ , we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac–Moody algebras.- Open Access
- 14 May 2018
- Published by SIGMA (Symmetry, Integrability and Geometry: Methods and Application) in Symmetry, Integrability and Geometry: Methods and Applications
- Vol. 14
A τ-Tilting Approach to Dissections of Polygons
- 14 May 2018
- Published by SIGMA (Symmetry, Integrability and Geometry: Methods and Application) in Symmetry, Integrability and Geometry: Methods and Applications
- Vol. 14
Open AccessA τ-Tilting Approach to Dissections of Polygons
- 12 April 2018
- Published by Oxford University Press (OUP) in International Mathematics Research Notices
- Vol. 2020 (3) , 914-956
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.- 12 April 2018
- Published by Oxford University Press (OUP) in International Mathematics Research Notices
- Vol. 2020 (3) , 914-956
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.- 1 April 2018
- Published by Elsevier in Advances in Mathematics
- Vol. 328, 713-749
Polytopal realizations of finite type g-vector fans
- 1 April 2018
- Published by Elsevier in Advances in Mathematics
- Vol. 328, 713-749
Polytopal realizations of finite type g-vector fans
- Open Access
Polytopal realizations of finite type $\mathbf{g}$-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 $\mathsf{Asso}_{\mathrm{un}}(\Gamma)$.Open AccessPolytopal realizations of finite type $\mathbf{g}$-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 $\mathsf{Asso}_{\mathrm{un}}(\Gamma)$. - Open Access
- 1 January 2018
- Published by Mathematical Sciences Publishers in Pacific Journal of Mathematics
- Vol. 293 (1) , 179-206
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-seedmutation 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.
- 1 January 2018
- Published by Mathematical Sciences Publishers in Pacific Journal of Mathematics
- Vol. 293 (1) , 179-206
Open AccessInitial-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-seedmutation 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.