arXiv subjects

Karin Baur

Publications and source records attributed to Karin Baur.

At least 19 recordsLinked to original sources

arXiv2026

Growth of infinite frieze patterns of affine type

We analyse the growth coefficients of infinite frieze patterns arising from cluster algebras using cluster modular groups and cluster categories. For a fixed cluster category of affine type, we prove that the collection of infinite frieze patterns given by both the homogeneous and non-homogeneous stable tubes all have the same growth coefficients. We also derive and verify an explicit formula for the $k$-th growth coefficient, expressed directly in terms of data from homogeneous tubes, or, alternatively, from appropriate elements of the corresponding cluster algebra.

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arXiv2024

Global dimensions of local geodesic ghor algebras

A ghor algebra is a path algebra with relations of a dimer quiver in a compact surface. We show that the global dimension of any cyclic localization of a geodesic ghor algebra on a genus $g \geq 1$ surface is bounded above by $2g+1$.This number coincides with the Krull dimension of the center of the ghor algebra. We further show that the bound is an equality if and only if the point of localization sits over the noetherian locus of the center.

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arXiv2024

A correspondence between additive and monoidal categorifications with application to Grassmannian cluster categories

Building on work of Derksen-Fei and Plamondon, we formulate a conjectural correspondence between additive and monoidal categorifications of cluster algebras, which reveals a new connection between the additive reachability conjecture and the multiplicative reachability conjecture. Evidence for this conjecture is provided by results on Grassmannian cluster algebras and categories in the tame types. Moreover, we give a construction of the generic kernels introduced by Hernandez and Leclerc for type $\mathbb{A}$ via the Grassmannian cluster categories. As an application of the correspondence, we construct rigid indecomposable modules and indecomposable non-rigid modules in Grassmannian cluster categories.

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arXiv2024

Infinite friezes of affine type D

In this article, we study infinite friezes arising from cluster categories of affine type $D$ and determine the growth coefficients for these friezes. We prove that for each affine type $D$, the friezes given by the tubes all have the same growth behavior.

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arXiv2024

A geometric model for the module category of a string algebra

In this paper, we give a geometric construction of string algebras and of their module categories. Our approach uses dissections of punctured Riemann surfaces with extra data at marked points, called labels. As an application, we give a classification of support tau-tilting modules in terms of arcs in such a tiled surface. In the case when the string algebra is gentle, we recover the classification given arXiv:2004.11136.

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arXiv2024

Flip graphs of coloured triangulations of convex polygons

A triangulation of a polygon is a subdivision of it into triangles, using diagonals between its vertices. Two different triangulations of a polygon can be related by a sequence of flips: a flip replaces a diagonal by the unique other diagonal in the quadrilateral it defines. In this paper, we study coloured triangulations and coloured flips. In this more general situation, it is no longer true that any two triangulations can be linked by a sequence of (coloured) flips. In this paper, we study the connected components of the coloured flip graphs of triangulations. The motivation for this is a result of Gravier and Payan proving that the Four-Colour Theorem is equivalent to the connectedness of the flip graph of 2-coloured triangulations.

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arXiv2024

A gluing operation for dimer quivers

In this article we introduce a gluing operation on dimer models. This allows us to construct dimer quivers on arbitrary surfaces. We study how the associated dimer and boundary algebras behave under the gluing and how to determine them from the gluing components. We also use this operation to construct homogeneous dimer quivers on annuli.

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arXiv2023

Real roots in the root system $\mathsf{T}_{2,p,q}$

Motivated by the recent advances in the categorification of the cluster structure on the coordinate rings of Grassmannians of $k$-subspaces in $n$-space, we investigate a particular construction of root systems of type $\mathsf{T}_{2,p,q}$, including the type $\mathsf{E}_n$. This construction generalizes Manin's ``hyperbolic construction'' of $\mathsf{E}_8$ and reveals a lot of otherwise hidden regularities in this family of root systems.

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arXiv2023

Rigid Indecomposable Modules in Grassmannian Cluster Categories

The coordinate ring of the Grassmannian variety of $k$-dimensional subspaces in $\mathbb{C}^n$ has a cluster algebra structure with Plücker relations giving rise to exchange relations. In this paper, we study indecomposable modules of the corresponding Grassmannian cluster categories ${\rm CM}(B_{k,n})$. Jensen, King, and Su have associated a Kac-Moody root system $J_{k,n}$ to ${\rm CM}(B_{k,n})$ and shown that in the finite types, rigid indecomposable modules correspond to roots. In general, the link between the category ${\rm CM}(B_{k,n})$ and the root system $J_{k,n}$ remains mysterious and it is an open question whether indecomposables always give roots. In this paper, we provide evidence for this association in the infinite types: we show that every indecomposable rank 2 module corresponds to a root of the associated root system. We also show that indecomposable rank 3 modules in ${\rm CM}(B_{3,n})$ all give rise to roots of $J_{3,n}$. For the rank 3 modules in ${\rm CM}(B_{3,n})$ corresponding to real roots, we show that their underlying profiles are cyclic permutations of a certain canonical one. We also characterize the rank 3 modules in ${\rm CM}(B_{3,n})$ corresponding to imaginary roots. By proving that there are exactly 225 profiles of rigid indecomposable rank 3 modules in ${\rm CM}(B_{3,9})$ we confirm the link between the Grassmannian cluster category and the associated root system in this case. We conjecture that the profile of any rigid indecomposable module in ${\rm CM}(B_{k,n})$ corresponding to a real root is a cyclic permutation of a canonical profile.

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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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arXiv2022

Infinite friezes and triangulations of annuli

It is known that any infinite frieze comes from a triangulation of an annulus by Baur, Parsons and Tschabold. In this paper we show that each periodic infinite frieze determines a triangulation of an annulus in essentially a unique way. Since each triangulation of an annulus determines a pair of friezes, we study such pairs and show how they determine each other. We study associated module categories and determine the growth coefficient of the pair of friezes in terms of modules as well as their quiddity sequences.

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arXiv2021

A generalization of cancellative dimer algebras to hyperbolic surfaces

We study a new class of quiver algebras on surfaces, called 'geodesic ghor algebras'. These algebras generalize cancellative dimer algebras on a torus to higher genus surfaces, where the relations come from perfect matchings rather than a potential. Although cancellative dimer algebras on a torus are noncommutative crepant resolutions, the center of any dimer algebra on a higher genus surface is just the polynomial ring in one variable, and so the center and surface are unrelated. In contrast, we establish a rich interplay between the central geometry of geodesic ghor algebras and the topology of the surface in which they are embedded. Furthermore, we show that noetherian central localizations of such algebras are endomorphism rings of modules over their centers.

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arXiv2021

Construction of Rank $2$ Indecomposable Modules in Grassmannian Cluster Categories

The category ${\rm CM}(B_{k,n}) $ of Cohen-Macaulay modules over a quotient $B_{k,n}$ of a preprojective algebra provides a categorification of the cluster algebra structure on the coordinate ring of the Grassmannian variety of $k$-dimensional subspaces in $\mathbb C^n$, \cite{JKS16}. Among the indecomposable modules in this category are the rank $1$ modules which are in bijection with $k$-subsets of $\{1,2,\dots,n\}$, and their explicit construction has been given by Jensen, King and Su. These are the building blocks of the category as any module in ${\rm CM}(B_{k,n}) $ can be filtered by them. In this paper we give an explicit construction of rank 2 modules. With this, we give all indecomposable rank 2 modules in the cases when $k=3$ and $k=4$. In particular, we cover the tame cases and go beyond them. We also characterise the modules among them which are uniquely determined by their filtrations. For $k\ge 4$, we exhibit infinite families of non-isomorphic rank 2 modules having the same filtration.

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arXiv2021

Examples of geodesic ghor algebras on hyperbolic surfaces

Cancellative dimer algebras on a torus have many nice algebraic and homological properties. However, these nice properties disappear for dimer algebras on higher genus surfaces. We consider a new class of quiver algebras on surfaces, called 'geodesic ghor algebras', that reduce to cancellative dimer algebras on a torus, yet continue to have nice properties on higher genus surfaces. These algebras exhibit a rich interplay between their central geometry and the topology of the surface. We show that (nontrivial) geodesic ghor algebras do in fact exist, and give explicit descriptions of their central geometry. This article serves a companion to the article 'A generalization of cancellative dimer algebras to hyperbolic surfaces', where the main statement is proven.

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arXiv2021

Frieze patterns of integers

The famous theorem of Conway and Coxeter on frieze patterns gave a geometric interpretation to integral friezes via triangulations of polygons. In this article, we review this result and show some of the development it has led to. The last decade has seen a lot of activities on friezes. One reason behind this is the connection to cluster combinatorics.

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arXiv2020

Cluster algebras generated by projective cluster variables

We introduce the notion of a lower bound cluster algebra generated by projective cluster variables as a polynomial ring over the initial cluster variables and the so-called projective cluster variables. We show that under an acyclicity assumption, the cluster algebra and the lower bound cluster algebra generated by projective cluster variables coincide. In this case we use our results to construct a basis for the cluster algebra. We also show that any coefficient-free cluster algebra of types $A_n$ or $\widetilde{A}_n$ is equal to the corresponding lower bound cluster algebra generated by projective cluster variables.

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arXiv2020

Friezes satisfying higher SL$_k$-determinants

In this article, we construct SL$_k$-friezes using Plücker coordinates, making use of the cluster structure on the homogeneous coordinate ring of the Grassmannian of $k$-spaces in $n$-space via the Plücker embedding. When this cluster algebra is of finite type, the SL$_k$-friezes are in bijection with the so-called mesh friezes of the corresponding Grassmannian cluster category. These are collections of positive integers on the AR-quiver of the category with relations inherited from the mesh relations on the category. In these finite type cases, many of the SL$_k$-friezes arise from specialising a cluster to 1. These are called unitary. We use Iyama-Yoshino reduction to analyse the non-unitary friezes. With this, we provide an explanation for all known friezes of this kind. An appendix by Cuntz and Plamondon proves that there are 868 friezes of type $E_6$.

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arXiv2019

Classification of cosilting modules in type $\tilde{A}$

Torsion pairs in the category of finitely presented modules over a noetherian ring can be parametrised by the class of cosilting modules. In this paper, we characterise such modules in terms of their indecomposable summands, providing a new approach to the classification of torsion pairs. In particular, we classify cosilting modules over cluster-tilted algebras of type $\tilde{A}$. We do this by using a geometric model for finite- and infinite-dimensional modules over such algebras.

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