arXiv subjects

Diego Velasco

Publications and source records attributed to Diego Velasco.

3 recordsLinked to original sources

arXiv2023

Orbifold diagrams

We study alternating strand diagrams on the disk with an orbifold point. These are quotients by rotation of Postnikov diagrams on the disk, and we call them orbifold diagrams. We associate a quiver with potential to each orbifold diagram, in such a way that its Jacobian algebra and the one associated to the covering Postnikov diagram are related by a skew-group algebra construction. We moreover realise this Jacobian algebra as the endomorphism algebra of a certain explicit cluster-tilting object. This is similar to (and relies on) a result by Baur-King-Marsh for Postnikov diagrams on the disk.

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arXiv2019

On a family of Caldero-Chapoton algebras that have the Laurent phenomenon

We realize a family of generalized cluster algebras as Caldero-Chapoton algebras of quivers with relations. Each member of this family arises from an unpunctured polygon with one orbifold point of order 3, and is realized as a Caldero-Chapoton algebra of a quiver with relations naturally associated to any triangulation of the alluded polygon. The realization is done by defining for every arc $j$ on the polygon with orbifold point a representation $M(j)$ of the referred quiver with relations, and by proving that for every triangulation $τ$ and every arc $j\inτ$, the product of the Caldero-Chapoton functions of $M(j)$ and $M(j')$, where $j'$ is the arc that replaces $j$ when we flip $j$ in $τ$, equals the corresponding exchange polynomial of Chekhov-Shapiro in the generalized cluster algebra. Furthermore, we show that there is a bijection between the set of generalized cluster variables and the isomorphism classes of $E$-rigid indecomposable decorated representations of $Λ$.

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arXiv2015

Mutation of representations and nearly Morita equivalence

Buan-Iyama-Reiten-Smith proved, based on Derksen-Weyman-Zelevinsky work, that the Jacobian algebra of two quivers with potential related by a QP-mutation are nearly Morita equivalent. They proved, using Axiom of Choice, that the natural functor $μ_k$ is an equivalence by showing that $μ_k$ is full, faithfull and dense. In this note we provide a quasi-inverse $μ_k^-$ to $μ_k$ without Axiom of Choice.

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