Searcharxiv⌕ Search

arXiv subjects

Pierre-Guy Plamondon

Publications and source records attributed to Pierre-Guy Plamondon.

At least 19 recordsLinked to original sources
arXiv2026

Configuration Spaces of Finite Representation Type Algebras

To every finite-dimensional $\mathbb C$-algebra $Λ$ of finite representation type we associate an affine variety. These varieties are a large generalization of the varieties defined by "$u$ variables" satisfying "$u$-equations", first introduced in the context of open string theory and moduli space of ordered points on the real projective line by Koba and Nielsen, rediscovered by Brown as "dihedral co-ordinates", and recently generalized to any finite type hereditary algebras. We show that each such variety is irreducible and admits a rational parametrization. The assignment is functorial: algebra quotients correspond to monomial maps among the varieties. The non-negative real part of each variety has boundary strata that are controlled by Jasso reduction. These non-negative parts naturally define a generalization of open string integrals in physics, exhibiting factorization and splitting properties that do not come from a worldsheet picture. We further establish a family of Rogers dilogarithm identities extending results of Chapoton beyond the Dynkin case.

math.RT↗
arXiv2026

Skew-group $A_{\infty}$-categories as Fukaya categories of orbifolds

We study the partially wrapped Fukaya category of a surface with boundary with an action of a group of order two. Inspired by skew-group algebras and categories, we define the notion of a skew-group $A_\infty$-category and let it play the role of the partially wrapped Fukaya category of an orbifold surface. We classify indecomposable objects in terms of graded curves with signs, or taggings, at orbifold points. We compute morphisms between a class of objects, and we use this to describe tilting objects and find algebras derived-equivalent to skew-gentle algebras.

math.RT↗
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.

math.CO↗
arXiv2025

A geometric model for the derived category of gentle algebras

In this paper we construct a geometric model for the triangulated category generated by the simple modules of any graded gentle algebra. This leads to a geometric model of their perfect derived categories and by a recent paper of Booth, Goodbody and the first author also of their derived categories of objects with finite-dimensional cohomology. The construction is based on the ribbon graph associated to a gentle algebra in the work of the third author, and is linked to partially wrapped Fukaya categories by the work of Haiden, Katzarkov and Kontsevich and to derived categories of coherent sheaves on nodal stacky curves by the work of Lekili and Polishchuk. The ribbon graph gives rise to an oriented surface with boundary and marked points in the boundary. We show that the homotopy classes of curves connecting marked points and of closed curves are in bijection with the isomorphism classes of indecomposable objects in the derived category of the graded gentle algebra. Intersections of curves correspond to morphisms and resolving the crossings of curves gives rise to mapping cones. The Auslander-Reiten translate corresponds to rotating endpoints of curves along the boundary. Furthermore, we show that the surface encodes the derived invariant of Avella-Alaminos and Geiss.

math.RT↗
arXiv2024

All Loop Scattering For All Multiplicity

This is part of a series of papers describing the new curve integral formalism for scattering amplitudes of the colored scalar tr$ϕ^3$ theory. We show that the curve integral manifests a very surprising fact about these amplitudes: the dependence on the number of particles, $n$, and the loop order, $L$, is effectively decoupled. We derive the curve integrals at tree-level for all $n$. We then show that, for higher loop-order, it suffices to study the curve integrals for $L$-loop tadpole-like amplitudes, which have just one particle per color trace-factor. By combining these tadpole-like formulas with the the tree-level result, we find formulas for the all $n$ amplitudes at $L$ loops. We illustrate this result by giving explicit curve integrals for all the amplitudes in the theory, including the non-planar amplitudes, through to two loops, for all $n$.

hep-th↗
arXiv2024

Extriangulated ideal quotients, with applications to cluster theory and gentle algebras

We extend results of Brüstle-Yang on ideal quotients of 2-term subcategories of perfect derived categories of non-positive dg algebras to a relative setting. We find a new interpretation of such quotients: they appear as prototypical examples of a new construction of quotients of extriangulated categories by ideals generated by morphisms from injectives to projectives. We apply our results to Frobenius exact cluster categories and Higgs categories with suitable relative extriangulated structures, and to categories of walks related to gentle algebras. In all three cases, the extriangulated structures are well-behaved (they are 0-Auslander) and their quotients are equivalent to homotopy categories of two-term complexes of projectives over suitable finite-dimensional algebras.

math.RT↗
arXiv2023

A refined multiplication formula for cluster characters

We obtain a multiplication formula for cluster characters on (stably) 2-Calabi-Yau (Frobenius or) triangulated categories. This formula generalizes those known for arbitrary pairs of objects and for Auslander-Reiten triangles. As an application, we show that for cluster algebras of acyclic types, specialization of a cluster variable to 1 sends all cluster variables to elements of a cluster algebra of smaller rank. We also obtain application to the reduction of friezes of acyclic type.

math.RT↗
arXiv2022

Non-kissing complexes and tau-tilting for gentle algebras

We interpret the support $τ$-tilting complex of any gentle bound quiver as the non-kissing complex of walks on its blossoming quiver. Particularly relevant examples were previously studied for quivers defined by a subset of the grid or by a dissection of a polygon. We then focus on the case when the non-kissing complex is finite. We show that the graph of increasing flips on its facets is the Hasse diagram of a congruence-uniform lattice. Finally, we study its $\mathbf{g}$-vector fan and prove that it is the normal fan of a non-kissing associahedron.

math.CO↗
arXiv2022

Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors

We show that the mesh mutations are the minimal relations among the $\boldsymbol{g}$-vectors with respect to any initial seed in any finite type cluster algebra. We then use this algebraic result to derive geometric properties of the $\boldsymbol{g}$-vector fan: we show that the space of all its polytopal realizations is a simplicial cone, and we then observe that this property implies that all its realizations can be described as the intersection of a high dimensional positive orthant with well-chosen affine spaces. This sheds a new light on and extends earlier results of N. Arkani-Hamed, Y. Bai, S. He, and G. Yan in type $A$ and of V. Bazier-Matte, G. Douville, K. Mousavand, H. Thomas and E. Yildirim for acyclic initial seeds. Moreover, we use a similar approach to study the space of polytopal realizations of the $\boldsymbol{g}$-vector fans of another generalization of the associahedron: non-kissing complexes (a.k.a. support $τ$-tilting complexes) of gentle algebras. We show that the space of realizations of the non-kissing fan is simplicial when the gentle bound quiver is brick and $2$-acyclic, and we describe in this case its facet-defining inequalities in terms of mesh mutations. Along the way, we prove algebraic results on $2$-Calabi-Yau triangulated categories, and on extriangulated categories that are of independent interest. In particular, we prove, in those two setups, an analogue of a result of M. Auslander on minimal relations for Grothendieck groups of module categories.

math.RT↗
arXiv2020

Tame algebras have dense $\mathbf{g}$-vector fans

The $\mathbf{g}$-vector fan of a finite-dimensional algebra is a fan whose rays are the $\mathbf{g}$-vectors of its $2$-term presilting objects. We prove that the $\mathbf{g}$-vector fan of a tame algebra is dense. We then apply this result to obtain a near classification of quivers for which the closure of the cluster $\mathbf{g}$-vector fan is dense or is a half-space, using the additive categorification of cluster algebras by means of Jacobian algebras. As another application, we prove that for quivers with potentials arising from once-punctured closed surfaces, the stability and cluster scattering diagrams only differ by wall-crossing functions on the walls contained in a separating hyperplane. The appendix is devoted to the construction of truncated twist functors and their adjoints.

math.RT↗
arXiv2019

Quivers with potentials and actions of finite abelian groups

Let $G$ be a finite abelian group acting on a path algebra $kQ$ by permuting the vertices and preserving the arrowspans. Let $W$ be a potential on the quiver $Q$ which is fixed by the action. We study the skew group dg algebra $Γ_{Q, W}G$ of the Ginzburg dg algebra of $(Q, W)$. It is known that $Γ_{Q, W}G$ is Morita equivalent to another Ginzburg dg algebra $Γ_{Q_G, W_G}$, whose quiver $Q_G$ was constructed by Demonet. In this article we give an explicit construction of the potential $W_G$ as a linear combination of cycles in $Q_G$, and write the Morita equivalence explicitly. As a corollary, we obtain functors between the cluster categories corresponding to the two quivers with potentials.

math.RT↗
arXiv2019

Unistructurality of cluster algebras from surfaces without punctures

A cluster algebra is unistructural if the set of its cluster variables determines its clusters and seeds. It is conjectured that all cluster algebras are unistructural. In this paper, we show that any cluster algebra arising from a triangulation of a marked surface without punctures is unistructural. Our proof relies on the existence of a positive basis known as the bracelet basis, and on the skein relations. We also prove that a cluster algebra defined from a disjoint union of quivers is unistructural if and only if the cluster algebras defined from the connected components of the quiver are unistructural.

math.RT↗
arXiv2019

A complete derived invariant for gentle algebras via winding numbers and Arf invariants

Gentle algebras are in bijection with admissible dissections of marked oriented surfaces. In this paper, we further study the properties of admissible dissections and we show that silting objects for gentle algebras are given by admissible dissections of the associated surface. We associate to each gentle algebra a line field on the corresponding surface and prove that the derived equivalence class of the algebra is completely determined by the homotopy class of the line field up to homeomorphism of the surface. Then, based on winding numbers and the Arf invariant of a certain quadratic form over $\mathbb Z_2$, we translate this to a numerical complete derived invariant for gentle algebras.

math.RT↗
arXiv2018

Non-kissing and non-crossing complexes for locally gentle algebras

Starting from a locally gentle bound quiver, we define on the one hand a simplicial complex, called the non-kissing complex. On the other hand, we construct a punctured, marked, oriented surface with boundary, endowed with a pair of dual dissections. From those geometric data, we define two simplicial complexes: the accordion complex, and the slalom complex, generalizing work of A. Garver and T. McConville in the case of a disk. We show that all three simplicial complexes are isomorphic, and that they are pure and thin. In particular, there is a notion of mutation on their facets, akin to $τ$-tilting mutation. Along the way, we also construct inverse bijections between the set of isomorphism classes of locally gentle bound quivers and the set of homeomorphism classes of punctured, marked, oriented surfaces with boundary, endowed with a pair of dual dissections.

math.CO↗
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.

math.RT↗
arXiv2018

The cluster category of a surface with punctures via group actions

Given a certain triangulation of a punctured surface with boundary, we construct a new triangulated surface without punctures which covers it. This new surface is naturally equipped with an action of a group of order two, and its quotient by this action recovers the original surface. We show that the group acts on the quivers with potentials associated to the surfaces, and that their Ginzburg dg algebras are skew group algebras of each other, up to Morita equivalence. We then use these results to construct functors between the generalized cluster categories associated to the triangulations. This allows us to give a complete description of the indecomposable objects of these categories in terms of curves on the surface, when the surface has punctures and non-empty boundary.

math.RT↗