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Salvatore Stella

Publications and source records attributed to Salvatore Stella.

At least 19 recordsLinked to original sources
arXiv2026

Flip Combinatorial Invariance and Weyl groups

In this work, we investigate the approach via flipclasses to the Combinatorial Invariance Conjecture for Kazhdan--Lusztig polynomials of all Coxeter groups. We prove the combinatorial invariance of Kazhdan--Lusztig $\widetilde{R}$-polynomials of Weyl groups modulo $q^7$ and of Kazhdan--Lusztig $\widetilde{R}$-polynomials of type $A$ Weyl groups modulo $q^8$. As a consequence, we establish the Combinatorial Invariance Conjecture for all intervals up to length 10 in Weyl groups and up to length 12 in type $A$ Weyl groups.

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arXiv2026

Mutation of theta functions

We give an account of mutation of theta functions in cluster scattering diagrams, starting with a notion of mutation that is related to, but different from, the notion of mutation defined by Gross, Hacking, Keel, and Kontsevich. This different approach to mutation leads to several applications. Three of the applications simplify the process of computing structure constants for multiplication of theta functions, and these are used in another paper on cluster scattering diagrams of affine type. Notable in these three applications is the appearance of mutation symmetries and dominance regions. The other two applications have to do with pointed reduced bases, a variation on the pointed bases of Fan Qin. We give a characterization of pointed reduced bases analogous to Qin's characterization of pointed bases. All of these applications take place in a version of Gross, Hacking, Keel, and Kontsevich's canonical algebra that can be constructed for an arbitrary exchange matrix.

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arXiv2026

Neighboring seeds in affine type: Universal coefficients and finite mutation-type

Neighboring seeds in a cluster algebra of affine type are seeds that are as close as possible to the boundary of the g-vector fan. This short note highlights the characterization of neighboring seeds given in a recent paper of Reading, Rupel, and Stella and applies that characterization in two ways. We prove a conjecture on the construction of universal geometric cluster algebras of affine type. We also characterize extended exchange matrices of affine type that are mutation-finite.

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arXiv2026

Dominance regions for affine cluster algebras

We determine dominance regions associated to cluster algebras of affine type. In the most interesting cases, the dominance region is a line segment, which we describe explicitly. Motivations for this work include a project to determine all pointed bases for cluster algebras of affine type and a separate application that determines all theta functions in the affine case. The proofs draw on known results from the doubled Cambrian fan and almost-positive roots models, as well as a new tool that we develop: a detailed description of neighboring seeds of affine type (seeds that are, in some sense, as close as possible to the boundary of the g-vector fan).

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arXiv2026

On the growth of friezes via theta functions

We prove that the infinite friezes arising from the tubes of a given cluster algebra of acyclic affine type all have the same growth coefficients. Our proof uses identities satisfied by theta functions. This generalizes previous results in affine types~$ADE$ by several groups of authors.

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arXiv2026

Theta functions in acyclic affine type

We characterize the theta functions for vectors in the imaginary wall in a cluster algebra of acyclic affine type and compute some of their structure constants. One of the structure constant computations can be interpreted as new "imaginary" exchange relations among cluster variables. We show that theta functions in the imaginary wall span a subalgebra of the cluster algebra that we call the imaginary subalgebra, which decomposes as a tensor product of tube subalgebras that are generalized cluster algebras of type C. Our proofs exploit mutation-symmetries of the exchange matrix, an earlier characterization of dominance regions in affine type, and combinatorial models for cluster scattering diagrams of acyclic affine type.

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arXiv2025

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.

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arXiv2020

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.

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arXiv2019

On the Strength of Chromatic Symmetric Homology for graphs

In this paper, we investigate the strength of chromatic symmetric homology as a graph invariant. Chromatic symmetric homology is a lift of the chromatic symmetric function for graphs to a homological setting, and its Frobenius characteristic is a q,t generalization of the chromatic symmetric function. We exhibit three pairs of graphs where each pair has the same chromatic symmetric function but distinct homology. We also show that integral chromatic symmetric homology contains torsion, and based on computations, conjecture that Z_2-torsion in bigrading (1,0) detects nonplanarity in the graph.

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arXiv2019

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 $Φ_c$ and a compatibility degree, given by a formula that is new even in finite type. The clusters define a complete fan $\operatorname{Fan}_c(Φ)$. Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of $\operatorname{Fan}_c(Φ)$ induced by real roots to the ${\mathbf g}$-vector fan of the associated cluster algebra. We show that $Φ_c$ 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.

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arXiv2019

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.

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arXiv2018

A $τ$-Tilting Approach to Dissections of Polygons

We show that any accordion complex associated to a dissection of a convex polygon is isomorphic to the support $τ$-tilting simplicial complex of an explicit finite dimensional algebra. To this end, we prove a property of some induced subcomplexes of support $τ$-tilting simplicial complexes of finite dimensional algebras.

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arXiv2018

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 $Γ$, we construct a universal associahedron $\mathsf{Asso}_{\mathrm{un}}(Γ)$ with the property that any $\mathbf{g}$-vector fan of type $Γ$ is the normal fan of a suitable projection of $\mathsf{Asso}_{\mathrm{un}}(Γ)$.

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arXiv2017

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 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.

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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.

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