arXiv subjects

Azzurra Ciliberti

Publications and source records attributed to Azzurra Ciliberti.

6 recordsLinked to original sources

arXiv2026

Skew-symmetrizable cluster algebras from surfaces and symmetric quivers

We study skew-symmetrizable cluster algebras $\mathcal{A}$ associated with unpunctured surfaces $\tilde{\mathbf{S}}$ endowed with an orientation-preserving involution $ς$. We give a geometric realization of such cluster algebras by showing that cluster variables of $\mathcal{A}$ correspond to non-crossing $ς$-orbits $[γ]$ of arcs of $\tilde{\mathbf{S}}$, while clusters are given by admissible $ς$-invariant triangulations. We establish a formula expressing some cluster variables of $\mathcal{A}$ in terms of those of a skew-symmetric cluster algebra of the same rank, which is combinatorially derived from $\mathcal{A}$. We use this result to provide a cluster expansion formula in terms of perfect matchings of some labeled modified snake graphs constructed from the arcs of $[γ]$. Then, we associate a symmetric finite-dimensional algebra $A$ to $\mathcal{A}$, such that non-initial cluster variables correspond to some orthogonal indecomposable $A$-modules. Finally, we exhibit a purely representation-theoretic map to $\mathcal{A}$, providing a Caldero-Chapoton map in this setting.

math.RT↗

arXiv2026

A categorification of cluster algebras of type B and C through symmetric quivers

We express cluster variables of type $B_n$ and $C_n$ in terms of cluster variables of type $A_n$. Then we associate a cluster tilted bound symmetric quiver $Q$ of type $A_{2n-1}$ to any seed of a cluster algebra of type $B_n$ and $C_n$. Under this correspondence, cluster variables of type $B_n$ (resp. $C_n$) correspond to orthogonal (resp. symplectic) indecomposable representations of $Q$. We find a Caldero-Chapoton map in this setting. We also give a categorical interpretation of the cluster expansion formula in the case of acyclic quivers.

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arXiv2025

A multiplication formula for cluster characters in gentle algebras

We prove a multiplication formula for cluster characters induced by generating extensions in a gentle algebra A, generalizing a result of Cerulli Irelli, Esposito, Franzen, Reineke. In the case where A is the gentle algebra of a triangulation T of an unpunctured marked surface, this provides a representation-theoretic interpretation of the exchange relations in the cluster algebra with principal coefficients in T. As an application, we interpret a formula that relates cluster variables of type B to cluster variables of type A in the symmetric module category of the algebras arising from special triangulations of a regular polygon.

math.RT↗

arXiv2025

Cluster expansion formulas and perfect matchings for type B and C

Let $\mathbf{P}_{2n+2}$ be the regular polygon with $2n+2$ vertices, and let $θ$ be the rotation of 180$^\circ$. Fomin and Zelevinsky proved that $θ$-invariant triangulations of $\mathbf{P}_{2n+2}$ are in bijection with the clusters of cluster algebras of type $B_n$ or $C_n$. Furthermore, cluster variables correspond to the orbits of the action of $θ$ on the diagonals of $\mathbf{P}_{2n+2}$. In this paper, we associate a labeled modified snake graph $\mathcal{G}_{ab}$ to each $θ$-orbit $[a,b]$, and we get the cluster variables of type $B_n$ and $C_n$ which correspond to $[a,b]$ as perfect matching Laurent polynomials of $\mathcal{G}_{ab}$. This extends the work of Musiker for cluster algebras of type B and C to every seed.

math.RT↗

arXiv2023

A deletion-contraction long exact sequence for chromatic symmetric homology

Crew and Spirklt generalize Stanley's chromatic symmetric function to vertex-weighted graphs. One of the primary motivations for extending the chromatic symmetric function to vertex-weighted graphs is the existence of a deletion-contraction relation in this setting, which, as known, holds for the chromatic polynomial, but doesn't hold for the chromatic symmetric function. In this paper we find a categorification of their new invariant extending the definition of chromatic symmetric homology to vertex-weighted graphs. We prove the existence of a deletion-contraction long exact sequence for chromatic symmetric homology which lifts the deletion-contraction relation that holds for the extension of Crew and Spirklt. Moreover, the new categorification gives a useful computational tool and allow us to answer two questions left open by Chandler, Sazdanovic, Stella and Yip. In particular, we prove that, for a graph G with $n$ vertices, the maximal index with nonzero homology is not greater that $n$ - 1. Moreover, we show that the homology is non-trivial for all the indices between the minimum and the maximum with this property.

math.CO↗

arXiv2022

On chromatic symmetric homology and planarity of graphs

Sazdanovic and Yip defined a categorification of Stanley's chromatic function called the chromatic symmetric homology. In this paper we prove that (as conjectured by Chandler, Sazdanovic, Stella and Yip), if a graph $G$ is non-planar, then its chromatic symmetric homology in bidegree (1,0) contains $\mathbb{Z}_2$-torsion. Our proof follows a recursive argument based on Kuratowsky's theorem.

math.CO↗