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Thorsten Holm

Publications and source records attributed to Thorsten Holm.

At least 19 recordsLinked to original sources
arXiv2026

Finite and infinite frieze patterns from p-angulations and a generalization of Weyl groupoids

A classic result of Conway and Coxeter on frieze patterns has been generalized to a bijection between $p$-angulations of regular polygons and frieze patterns of type $\Lambda_p$. One of the features of Conway-Coxeter theory is a combinatorial procedure to obtain from the triangulation all entries of the corresponding frieze pattern. We first present a combinatorial algorithm, involving Chebyshev polynomials, for obtaining from a dissection all entries of the corresponding frieze pattern. As an application we obtain a characterisation of frieze patterns of types $\Lambda_4$ and $\Lambda_6$ in terms of all entries (not only the quiddity cycle). We then study infinite frieze patterns of type $\Lambda_p$, which appeared in a preprint by Banaian and Chen, generalizing the infinite frieze patterns of positive integers studied by Baur, Parsons and Tschabold. As our main result we obtain a combinatorial model for infinite frieze patterns of type $\Lambda_p$, these are in bijection with certain $p$-angulations of an infinite strip. This extends results by Baur, Parsons and Tschabold from $p=3$ to arbitrary $p\ge 3$, and also provides new insight in the classic case. Infinite frieze patterns of positive integers appear in the context of Weyl groupoids. In the final section we extend this to infinite frieze patterns of type $\Lambda_p$ for any $p\ge 3$ by introducing a generalization of Cartan graphs and Weyl groupoids. We show that, up to equivalence, there is a 1-1 correspondence between connected simply connected Cartan graphs of type $\Lambda_p$ of rank two with infinitely many vertices permitting a root system and infinite frieze patterns of type $\Lambda_p$.

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arXiv2025

The Plesken Lie algebra for associative algebras with anti-involution: semisimple cellular algebras

Cohen and Taylor, following an idea of Plesken, introduced a Lie algebra to the complex group algebra of a finite group and determined its structure, based on the character theory of the group. We show how the definition of this Plesken Lie algebra can be extended to any associative algebra with an anti-involution. After some examples we consider semisimple cellular algebras and prove that their Plesken Lie algebras are direct sums of orthogonal Lie algebras, the sizes of which are determined by the dimensions of the cell modules of the cellular algebra.

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arXiv2024

Non-commutative friezes and their determinants, the non-commutative Laurent phenomenon for weak friezes, and frieze gluing

This paper studies a non-commutative generalisation of Coxeter friezes due to Berenstein and Retakh. It generalises several earlier results to this situation: A formula for frieze determinants, a $T$-path formula expressing the Laurent phenomenon, and results on gluing friezes together. One of our tools is a non-commutative version of the weak friezes introduced by Canakci and Jorgensen.

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arXiv2024

Noncommutative frieze patterns with coefficients

Based on Berenstein and Retakh's notion of noncommutative polygons we introduce and study noncommutative frieze patterns. We generalize several notions and fundamental properties from the classic (commutative) frieze patterns to noncommutative frieze patterns, e.g. propagation formulae and $\mu$-matrices, quiddity cycles and reduction formulae, and we show that local noncommutative exchange relations and local triangle relations imply all noncommutative exchange relations and triangle relations. Throughout, we allow coefficients, so we obtain generalizations of results from our earlier paper on frieze patterns with coefficients from the commutative to the noncommutative setting.

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arXiv2023

Frieze patterns over algebraic numbers

Conway and Coxeter have shown that frieze patterns over positive rational integers are in bijection with triangulations of polygons. An investigation of frieze patterns over other subsets of the complex numbers has recently been initiated by Jorgensen and the first two authors. In this paper we first show that a ring of algebraic numbers has finitely many units if and only if it is an order in a quadratic number field $\mathbb{Q}(\sqrt{d})$ where $d<0$. We conclude that these are exactly the rings of algebraic numbers over which there are finitely many non-zero frieze patterns for any given height. We then show that apart from the cases $d\in \{-1,-2,-3,-7,-11\}$ all non-zero frieze patterns over the rings of integers $\mathcal{O}_d$ for $d<0$ have only integral entries and hence are known as (twisted) Conway-Coxeter frieze patterns.

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arXiv2022

Weak friezes and frieze pattern determinants

Frieze patterns have been introduced by Coxeter in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Canakci and Jorgensen, are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals. To any frieze pattern one can associate a symmetric matrix using a triangular fundamental domain of the frieze pattern in the upper and lower half of the matrix and putting zeroes on the diagonal. Broline, Crowe and Isaacs have found a formula for the determinants of these matrices and their work has later been generalized in various directions by other authors. These frieze pattern determinants are the main focus of our paper. As our main result we show that this determinant behaves well with respect to gluing weak friezes: the determinant is the product of the determinants for the pieces glued, up to a scalar factor coming from the gluing diagonal. Then we give several applications of this result, showing that formulas from the literature, obtained by Broline-Crowe-Isaacs, Baur-Marsh, Bessenrodt-Holm-Jorgensen and Maldonado can all be obtained as consequences of our result.

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arXiv2020

Virtual mutations of weighted surface algebras

The finite-dimensional symmetric algebras over an algebraically closed field, based on surface triangulations, motivated by the theory of cluster algebras, have been extensively investigated and applied. In particular, the weighted surface algebras and their deformations were introduced and studied in [16]-[20], and it was shown that all these algebras, except few singular cases, are symmetric tame periodic algebras of period $4$. In this article, using the general form of a weighted surface algebra from [19], we introduce and study so called virtual mutations of weighted surface algebras, which constitute a new large class of symmetric tame periodic algebras of period $4$. We prove that all these algebras are derived equivalent but not isomorphic to weighted surface algebras. We associate such algebras to any triangulated surface, first taking blow-ups of a family of edges to $2$-triangle discs, and then virtual mutations of their weighted surface algebras. The results of this paper form an essential step towards a classification of all tame symmetric periodic algebras.

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arXiv2020

Subpolygons in Conway-Coxeter frieze patterns

Friezes with coefficients are maps assigning numbers to the edges and diagonals of a regular polygon such that all Ptolemy relations for crossing diagonals are satisfied. Among these, the classic Conway-Coxeter friezes are the ones where all values are natural numbers and all edges have value 1. Every subpolygon of a Conway-Coxeter frieze yields a frieze with coefficients over the natural numbers. In this paper we give a complete arithmetic criterion for which friezes with coefficients appear as subpolygons of Conway-Coxeter friezes. This generalizes a result of our earlier paper with Peter Jorgensen from triangles to subpolygons of arbitrary size.

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arXiv2019

Frieze patterns with coefficients

Frieze patterns, as introduced by Coxeter in the 1970's, are closely related to cluster algebras without coefficients. A suitable generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp. In this paper we study these frieze patterns with coefficients systematically and prove various fundamental results, generalizing classic results for frieze patterns. As a consequence we see how frieze patterns with coefficients can be obtained from classic frieze patterns by cutting out subpolygons from the triangulated polygons associated to classic Conway-Coxeter frieze patterns. We address the question of which frieze patterns with coefficients can be obtained in this way and solve this problem completely for triangles. Finally, we prove a finiteness result for frieze patterns with coefficients by showing that for a given boundary sequence there are only finitely many (non-zero) frieze patterns with coefficients with entries in a discrete subset of the complex numbers.

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arXiv2018

Frieze patterns over integers and other subsets of the complex numbers

We study (tame) frieze patterns over subsets of the complex numbers, with particular emphasis on the corresponding quiddity cycles. We provide new general transformations for quiddity cycles of frieze patterns. As one application, we present a combinatorial model for obtaining the quiddity cycles of all tame frieze patterns over the integers (with zero entries allowed), generalising the classic Conway-Coxeter theory. This model is thus also a model for the set of specializations of cluster algebras of Dynkin type $A$ in which all cluster variables are integers. Moreover, we address the question of whether for a given height there are only finitely many non-zero frieze patterns over a given subset $R$ of the complex numbers. Under certain conditions on $R$, we show upper bounds for the absolute values of entries in the quiddity cycles. As a consequence, we obtain that if $R$ is a discrete subset of the complex numbers then for every height there are only finitely many non-zero frieze patterns over $R$. Using this, we disprove a conjecture of Fontaine, by showing that for a complex $d$-th root of unity $ζ_d$ there are only finitely many non-zero frieze patterns for a given height over $R=\mathbb{Z}[ζ_d]$ if and only if $d\in \{1,2,3,4,6\}$.

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arXiv2018

A $p$-angulated generalisation of Conway and Coxeter's theorem on frieze patterns

Coxeter defined the notion of frieze pattern, and Conway and Coxeter proved that triangulations of polygons are in bijection with integral frieze patterns. We show a $p$-angulated generalisation involving non-integral frieze patterns. We also show that polygon dissections give rise to even more general non-integral frieze patterns.

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arXiv2017

All $SL_2$-tilings come from infinite triangulations

An $SL_2$-tiling is a bi-infinite matrix of positive integers such that each adjacent 2 by 2 submatrix has determinant 1. Such tilings are infinite analogues of Conway-Coxeter friezes, and they have strong links to cluster algebras, combinatorics, mathematical physics, and representation theory. We show that, by means of so-called Conway-Coxeter counting, every $SL_2$-tiling arises from a triangulation of the disc with two, three or four accumulation points. This improves earlier results which only discovered $SL_2$-tilings with infinitely many entries equal to 1. Indeed, our methods show that there are large classes of tilings with only finitely many entries equal to 1, including a class of tilings with no 1's at all. In the latter case, we show that the minimal entry of a tiling is unique.

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arXiv2015

Generalised friezes and a modified Caldero-Chapoton map depending on a rigid object, II

It is an important aspect of cluster theory that cluster categories are "categorifications" of cluster algebras. This is expressed formally by the (original) Caldero-Chapoton map X which sends certain objects of cluster categories to elements of cluster algebras. Let τc --> b --> c be an Auslander-Reiten triangle. The map X has the salient property that X(τc)X(c) - X(b) = 1. This is part of the definition of a so-called frieze. The construction of X depends on a cluster tilting object. In a previous paper, we introduced a modified Caldero-Chapoton map ρdepending on a rigid object; these are more general than cluster tilting objects. The map ρsends objects of sufficiently nice triangulated categories to integers and has the key property that ρ(τc)ρ(c) - ρ(b) is 0 or 1. This is part of the definition of what we call a generalised frieze. Here we develop the theory further by constructing a modified Caldero-Chapoton map, still depending on a rigid object, which sends objects of sufficiently nice triangulated categories to elements of a commutative ring A. We derive conditions under which the map is a generalised frieze, and show how the conditions can be satisfied if A is a Laurent polynomial ring over the integers. The new map is a proper generalisation of the maps X and ρ.

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arXiv2014

Generalized frieze pattern determinants and higher angulations of polygons

Frieze patterns (in the sense of Conway and Coxeter) are in close connection to triangulations of polygons. Broline, Crowe and Isaacs have assigned a symmetric matrix to each polygon triangulation and computed the determinant. In this paper we consider d-angulations of polygons and generalize the combinatorial algorithm for computing the entries in the associated symmetric matrices; we compute their determinants and the Smith normal forms. It turns out that both are independent of the particular d-angulation, the determinant is a power of d-1, and the elementary divisors only take values d-1 and 1. We also show that in the generalized frieze patterns obtained in our setting every adjacent 2x2-determinant is 0 or 1, and we give a combinatorial criterion for when they are 1, which in the case d=3 gives back the Conway-Coxeter condition on frieze patterns.

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arXiv2014

Cluster tilting vs. weak cluster tilting in Dynkin type A infinity

This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category C with the following properties. On the one hand, the d-cluster tilting subcategories of C have very simple mutation behaviour: Each indecomposable object has exactly d mutations. On the other hand, the weakly d-cluster tilting subcategories of C which lack functorial finiteness can have much more complicated mutation behaviour: For each 0 <= \ell <= d-1, we show a weakly d-cluster tilting subcategory T_{\ell} which has an indecomposable object with precisely \ell mutations. The category C is the algebraic triangulated category generated by a (d+1)-spherical object and can be thought of as a higher cluster category of Dynkin type A infinity.

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arXiv2014

Generalised friezes and a modified Caldero-Chapoton map depending on a rigid object

The (usual) Caldero-Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps "reachable" indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalises the idea that the cluster category is a "categorification" of the cluster algebra. The definition of the Caldero-Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster tilting object in the category. We study a modified version of the Caldero-Chapoton map which only requires the category to have a Serre functor, and only depends on a rigid object in the category. It is well-known that the usual Caldero-Chapoton map gives rise to so-called friezes, for instance Conway-Coxeter friezes. We show that the modified Caldero-Chapoton map gives rise to what we call generalised friezes, and that for cluster categories of Dynkin type A, it recovers the generalised friezes introduced by combinatorial means by Bessenrodt and us.

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arXiv2013

Towards derived equivalence classification of the cluster-tilted algebras of Dynkin type D

We provide a far reaching derived equivalence classification of the cluster-tilted algebras of Dynkin type D and suggest standard forms for the derived equivalence classes. We believe that the classification is complete, but some subtle questions remain open. We introduce another notion of equivalence called good mutation equivalence which is slightly stronger than derived equivalence but is algorithmically more tractable, and give a complete classification together with standard forms.

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