arXiv subjects

Karin Baur

Publications and source records attributed to Karin Baur.

At least 37 records · Page 2Linked to original sources

arXiv2019

Transformed flips in triangulations and matchings

Plane perfect matchings of $2n$ points in convex position are in bijection with triangulations of convex polygons of size $n+2$. Edge flips are a classic operation to perform local changes both structures have in common. In this work, we use the explicit bijection from Aichholzer et al. (2018) to determine the effect of an edge flip on the one side of the bijection to the other side, that is, we show how the two different types of edge flips are related. Moreover, we give an algebraic interpretation of the flip graph of triangulations in terms of elements of the corresponding Temperley-Lieb algebra.

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arXiv2019

Higher extensions for gentle algebras

In this paper we determine extensions of higher degree between indecomposable modules over gentle algebras. In particular, our results show how such extensions either eventually vanish or become periodic. We give a geometric interpretation of vanishing and periodicity of higher extensions in terms of the surface underlying the gentle algebra. For gentle algebras arising from triangulations of surfaces, we give an explicit basis of higher extension spaces between indecomposable modules.

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arXiv2018

Cluster categories from Grassmannians and root combinatorics

The category of Cohen-Macaulay modules of an algebra $B_{k,n}$ is used [JKS16] to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$-planes in $n$-space. In this paper, we find canonical Auslander--Reiten sequences and study the Auslander--Reiten translation periodicity for this category. Furthermore, we give an explicit construction of Cohen-Macaulay modules of arbitrary rank. We then use our results to establish a correspondence between rigid indecomposable modules of rank 2 and real roots of degree 2 for the associated Kac-Moody algebra in the tame cases.

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arXiv2018

Perfect k-colored matchings and (k+2)-gonal tilings

We derive a simple bijection between geometric plane perfect matchings on $2n$ points in convex position and triangulations on $n+2$ points in convex position. We then extend this bijection to monochromatic plane perfect matchings on periodically $k$-colored vertices and $(k+2)$-gonal tilings of convex point sets. These structures are related to a generalization of Temperley-Lieb algebras and our bijections provide explicit one-to-one relations between matchings and tilings. Moreover, for a given element of one class, the corresponding element of the other class can be computed in linear time.

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arXiv2018

Conway-Coxeter friezes and mutation: a survey

In this survey article we explain the intricate links between Conway-Coxeter friezes and cluster combinatorics. More precisely, we provide a formula, relying solely on the shape of the frieze, describing how each individual entry in the frieze changes under cluster mutation. Moreover, we provide a combinatorial formula for the number of submodules of a string module, and with that a simple way to compute the frieze associated to a fixed cluster tilting object in a cluster category of Dynkin type $A$ in the sense of Caldero and Chapoton.

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arXiv2018

Strongness of companion bases for cluster-tilted algebras of finite type

For every cluster-tilted algebra of simply-laced Dynkin type we provide a companion basis which is strong, i.e. gives the set of dimension vectors of the finitely generated indecomposable modules for the cluster-tilted algebra. This shows in particular that every companion basis of a cluster-tilted algebra of simply-laced Dynkin type is strong. Thus we give a proof of Parsons's conjecture.

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arXiv2018

A geometric model for the module category of a gentle algebra

In this article, gentle algebras are realised as tiling algebras, which are associated to partial triangulations of unpunctured surfaces with marked points on the boundary. This notion of tiling algebras generalise the notion of Jacobian algebras of triangulations of surfaces and the notion of surface algebras. We use this description to give a geometric model of the module category of any gentle algebra.

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arXiv2017

A generalised Euler-Poincaré formula for associahedra

We derive a formula for the number of flip-equivalence classes of tilings of an $n$-gon by collections of tiles of shape dictated by an integer partition $λ$. The proof uses the Euler-Poincaré formula; and the formula itself generalises the Euler-Poincaré formula for associahedra.

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arXiv2017

The fibres of the Scott map on polygon tilings are the flip equivalence classes

We associate strand diagrams to tilings of surfaces with marked points, generalising Scott's method for triangulations of polygons. We thus obtain a map from tilings of surfaces to permutations of the marked points on boundary components, the {\em Scott map}. In the disk case (polygon tilings) we prove that the fibres of the Scott map are the flip equivalence classes. The result allows us to consider the size of the image as a generalisation of a classical combinatorial problem, and hence to determine the size in low ranks.

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arXiv2017

Transfinite mutations in the completed infinity-gon

We introduce mutation along infinite admissible sequences for infinitely marked surfaces, that is surfaces with infinitely many marked points on the boundary. We show that mutation along such admissible sequences produces a preorder on the set of triangulations of a fixed infinitely marked surface. We provide a complete classification of the strong mutation equivalence classes of triangulations of the infinity- gon and the completed infinity-gon respectively, where strong mutation equivalence is the equivalence relation induced by this preorder. Finally, we introduce the notion of transfinite mutations in the completed infinity-gon and show that all its triangulations are transfinitely mutation equivalent, that is we can reach any triangulation of the completed infinity-gon from any other triangulation via a transfinite mutation.

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arXiv2017

Mutation of friezes

We study mutations of Conway-Coxeter friezes which are compatible with mutations of cluster-tilting objects in the associated cluster category of Dynkin type $A$. More precisely, we provide a formula, relying solely on the shape of the frieze, describing how each individual entry in the frieze changes under cluster mutation. We observe how the frieze can be divided into four distinct regions, relative to the entry at which we want to mutate, where any two entries in the same region obey the same mutation rule. Moreover, we provide a combinatorial formula for the number of submodules of a string module, and with that a simple way to compute the frieze associated to a fixed cluster-tilting object in a cluster category of Dynkin type $A$ in the sense of Caldero and Chapoton.

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arXiv2016

Growth behaviour of periodic tame friezes

We examine the growth behaviour of the entries occurring in $n$-periodic tame friezes of real numbers. Extending \cite{T}, we prove that generalised recursive relations exist between all entries of such friezes. These recursions are parametrised by a sequence of so-called growth coefficients, which are shown to satisfy itself a recursive relation. Thus, all growth coefficients are determined by a \emph{principle growth coefficients}, which can be read off directly from the frieze. We place special emphasis on periodic tame friezes of positive integers, specifying the values the growth coefficients take for any such frieze. We establish that the growth coefficients of the pair of friezes arising from a triangulation of an annulus coincide. The entries of both are shown to grow asymptotically exponentially, while triangulations of a punctured disc are seen to provide the only friezes of linear growth.

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arXiv2016

Dimer models and cluster categories of Grassmannians

We associate a dimer algebra A to a Postnikov diagram D (in a disk) corresponding to a cluster of minors in the cluster structure of the Grassmannian Gr(k,n). We show that A is isomorphic to the endomorphism algebra of a corresponding Cohen-Macaulay module T over the algebra B used to categorify the cluster structure of Gr(k,n) by Jensen-King-Su. It follows that B can be realised as the boundary algebra of A, that is, the subalgebra eAe for an idempotent e corresponding to the boundary of the disk. The construction and proof uses an interpretation of the diagram D, with its associated plabic graph and dual quiver (with faces), as a dimer model with boundary. We also discuss the general surface case, in particular computing boundary algebras associated to the annulus.

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arXiv2016

Extensions between Cohen-Macaulay modules of Grassmannian cluster categories

In this paper we study extensions between Cohen-Macaulay modules for algebras arising in the categorifications of Grassmannian cluster algebras. We prove that rank 1 modules are periodic, and we give explicit formulas for the computation of the period based solely on the rim of the rank 1 module in question. We determine ${\rm Ext}^i(L_I, L_J)$ for arbitrary rank 1 modules $L_I$ and $L_J$. An explicit combinatorial algorithm is given for computation of ${\rm Ext}^i(L_I, L_J)$ when $i$ is odd, and for $i$ even, we show that ${\rm Ext}^i(L_I, L_J)$ is cyclic over the centre, and we give an explicit formula for its computation. At the end of the paper we give a vanishing condition of ${\rm Ext}^i(L_I, L_J)$ for any $i>0$.

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arXiv2015

Infinite friezes

We provide a characterization of infinite frieze patterns of positive integers via triangulations of an infinite strip in the plane. In the periodic case, these triangulations may be considered as triangulations of annuli. We also give a geometric interpretation of all entries of infinite friezes via matching numbers.

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arXiv2015

A geometric realization of tame categories

We give a geometric realization of module categories of type $\tilde{A}_n$. We work with oriented arcs to define a translation quiver isomorphic to the Auslander-Reiten quiver of the module category of type $\tilde{A}_n$. To get a description of the module category, we introduce long moves between arcs. These allow us to include the infinite radical in the geometric description. Finally, our results can also be used to describe the corresponding cluster categories by taking unoriented arcs instead.

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arXiv2013

The quivers of the hereditary algebras of type $\tilde{A}_n$

This is a translation of the diploma thesis of Thomas Brüstle, Darstellungsköcher der erblichen Algebren vom Typ $\tilde{A}_n$, written in German, at the University of Zurich, August 1990. Supervisor: P. Gabriel. Translated by K. Baur, May 2013

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arXiv2013

Compactifying Exchange Graphs I: Annuli and Tubes

We introduce the notion of an \emph{asymptotic triangulation} of the annulus. We show that asymptotic triangulations can be mutated as the usual triangulations and describe their exchange graph. Viewing asymptotic triangulations as limits of triangulations under the action of the mapping class group, we compactify the exchange graph of the triangulations of the annulus. The cases of tubes are also considered.

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