arXiv subjects

Gordana Todorov

Publications and source records attributed to Gordana Todorov.

At least 19 recordsLinked to original sources

arXiv2026

Balanced notation for $τ$-rigid pairs

We introduce a new notation for $τ$-rigid pairs called ``balanced pairs''. The notation $\frac AB$ allows for simplification of some formulas, for example the Jasso category and its dual. Using this balanced notation we show that the $g$-vector $g(\frac AB)$ has nice geometric and algebraic implications: It defines ``lower'' and ``upper'' chambers and we prove that this corresponds to generalized Bongartz and co-Bongartz completions of balanced pairs. We also show that the balanced notation agrees the wall labels for the semi-invariant picture in the hereditary case and, in the general case, we describe the relation between balanced notation and the wall labels.

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arXiv2026

Short history of signed exceptional sequences

Whereas exceptional sequences have a long history with many well-known connections to combinatorics, signed exceptional sequences are relatively recent. The authors introduced this concept in 2017 [19], although it was retroactively realized that the category of noncrossing partitions [24] is a special case of this construction. Buan and Marsh [4] have introduced the concept of $τ$-exceptional sequences to generalize the definitions and theorems to all finite dimensional algebras. This short paper is the story of the original concept of signed exceptional sequences for hereditary algebras and how it developed out of the two authors' study of algebraic K-theory, link invariants, and cluster combinatorics.

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arXiv2025

Representation theory of hereditary artin algebras of finite representation type

Let $H$ be a hereditary artin algebra of finite representation type. We first determine all hammocks in the Auslander-Reiten quiver $\GaH$ of $\mmod H$, the category of finitely generated left $H$-modules. This enables us to obtain an effective method to construct $\GaH$ by simply viewing the ext-quiver of $H$. As easy applications, we compute the numbers of non-isomorphic indecomposable objects in $\mmod H$ and the associated cluster category $\mathscr{C}_H$, as well as the nilpotencies of the radicals of $\mmod H\hspace{-.4pt},$ $\hspace{-.5pt} D^{\hspace{.5pt}b\hspace{-.6pt}}(\hspace{-.5pt}\mmod H\hspace{-.5pt})$ and $\mathscr{C}_H$.

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arXiv2025

Picture groups and maximal green sequences

We show that picture groups are directly related to maximal green sequences for valued Dynkin quivers of finite type. Namely, there is a bijection between maximal green sequences and positive expressions (words in the generators without inverses) for the Coxeter element of the picture group. We actually prove the theorem for the more general set up of "vertically and horizontally ordered" sets of positive real Schur roots for any hereditary algebra (not necessarily of finite type). Furthermore, we show that every picture for such a set of positive roots is a linear combination of "atoms" and we give a precise description of atoms as special semi-invariant pictures.

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arXiv2024

Infinitesimal semi-invariant pictures and co-amalgamation

The purpose of this paper is to study the local structure of the semi-invariant picture of a tame hereditary algebra near the null root. Using a construction that we call co-amalgamation, we show that this local structure is completely described by the semi-invariant pictures of a collection of self-injective Nakayama algebras. We then describe the cones of this local structure using cluster-like structures that we call support regular clusters. Finally, we show that the local structure is (piecewise linearly) invariant under cluster tilting.

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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

Defect Invariant Nakayama Algebras

We show that for a given Nakayama algebra $Θ$, there exist countably many cyclic Nakayama algebras $Λ_i$, where $i \in \mathbb{N}$, such that the syzygy filtered algebra of $Λ_i$ is isomorphic to $Θ$ and we describe those algebras $Λ_i$. We show, among these algebras, there exists a unique algebra $Λ$ where the defects, representing the number of indecomposable injective but not projective modules, remain invariant for both $Θ$ and $Λ$. As an application, we achieve the classification of cyclic Nakayama algebras that are minimal Auslander-Gorenstein and dominant Auslander-regular algebras of global dimension three. Specifically, by using the Auslander-Iyama correspondence, we obtain cluster-tilting objects for certain Nakayama algebras. Additionally, we introduce cosyzygy filtered algebras and show that it is dual of syzygy filtered algebra.

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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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arXiv2022

Which cluster morphism categories are CAT(0)

The cluster morphism category of an hereditary algebra was introduced in [5] to show that the picture space of an hereditary algebra of finite representation type is a $K(π,1)$ for the associated picture group, thereby allowing for the computation of the homology of picture groups of finite type as carried out in [7] for the case of $A_n$. In this paper we show that the cluster morphism category is a $CAT(0)$-category for hereditary algebras of finite or tame type with only small tubes. As a consequence, we get that the classifying space of the cluster morphism category is a locally $CAT(0)$ space and, as a consequence of that, we get that this classifying space is a $K(π,1)$.

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arXiv2021

Continuous Quivers of Type A (III) Embeddings of Cluster Theories

We continue the work started in parts (I) and (II). In this part we classify which continuous type A quivers are derived equivalent and introduce the new continuous cluster category with E-clusters, which are a generalization of clusters. In the middle we provide a rigorous connection between the previous construction of the continuous cluster category and the new construction. We conclude with the introduction of a cluster theory, generalizing the notion of a cluster structure. Using this new notion, we demonstrate how one embeds known type A cluster theories into the new E-cluster theory in a way compatible with mutation. This is part (III) in a series of work that will conclude with a continuous generalization of mutation for cluster theories.

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arXiv2021

Continuous quivers of type A (I) Foundations

We generalize type $A$ quivers to continuous type $A$ quivers and prove initial results about pointwise finite-dimensional (pwf) representations. We classify the indecomosable pwf representations and provide a decomposition theorem, recovering results of Botnan and Crawley-Boevey. We also classify the indecomposable pwf projective representations. Finally, we prove that many of the properties of finite-dimensional type $A_n$ representations are present in finitely generated pwf representations. This is the self-contained foundational part of a series of works to study a generalization of continuous clusters categories and their relationship to other type $A$ cluster structures.

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arXiv2020

Dynamical Combinatorics and Torsion Classes

For finite semidistributive lattices the map $κ$ gives a bijection between the sets of completely join-irreducible elements and completely meet-irreducible elements. Here we study the $κ$-map in the context of torsion classes. It is well-known that the lattice of torsion classes for an artin algebra is semidistributive, but in general it is far from finite. We show the $κ$-map is well-defined on the set of completely join-irreducible elements, even when the lattice of torsion classes is infinite. We then extend $κ$ to a map on torsion classes which have canonical join representations given by the special torsion classes associated to the minimal extending modules introduced by the first and third authors and A. Carroll. For hereditary algebras, we show that the extended $κ$-map on torsion classes is essentially the same as Ringel's $ε$-map on wide subcategories. Also in hereditary case, we relate the square of $κ$ to the Auslander-Reiten translation.

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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

Continuous cluster categories II: continuous cluster-tilted categories

We show that the quotient of the continuous cluster category $\mathcal C_π$ modulo the additive subcategory generated by any cluster is an abelian category and we show that it is isomorphic to the category of infinite length modules over the endomorphism ring of the cluster. These theorems extend the theorems of Caldero-Chapoton-Schiffler and Buan-Marsh-Reiten for cluster categories to the continuous cluster category of type $A$. These results will be generalized in a series of forthcoming joint papers of the two authors with Job Rock.

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arXiv2019

Cyclic posets and triangulation clusters

Triangulated categories coming from cyclic posets were originally introduced by the authors in [IT15b] as a generalization of the constructions of various triangulated categories with cluster structures. We give an overview, then analyze triangulation clusters which are those corresponding to topological triangulations of the 2-disk. Locally finite non-triangulation clusters give topological triangulations of the cactus space associated to the cactus cyclic poset.

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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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arXiv2017

Dominant dimension and tilting modules

We study which algebras have tilting modules that are both generated and cogenerated by projective-injective modules. Crawley-Boevey and Sauter have shown that Auslander algebras have such tilting modules; and for algebras of global dimension $2$, Auslander algebras are classified by the existence of such tilting modules. In this paper, we show that the existence of such a tilting module is equivalent to the algebra having dominant dimension at least $2$, independent of its global dimension. In general such a tilting module is not necessarily cotilting. Here, we show that the algebras which have a tilting-cotilting module generated-cogenerated by projective-injective modules are precisely $1$-Auslander-Gorenstein algebras. When considering such a tilting module, without the assumption that it is cotilting, we study the global dimension of its endomorphism algebra, and discuss a connection with the Finitistic Dimension Conjecture. Furthermore, as special cases, we show that triangular matrix algebras obtained from Auslander algebras and certain injective modules, have such a tilting module. We also give a description of which Nakayama algebras have such a tilting module.

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arXiv2017

Signed exceptional sequences and the cluster morphism category

We introduce signed exceptional sequences as factorizations of morphisms in the cluster morphism category. The objects of this category are wide subcategories of the module category of a hereditary algebra. A morphism $[T]:\mathcal A\to \mathcal B$ is the equivalence class of a rigid object $T$ in the cluster category of $\mathcal A$ so that $\mathcal B$ is the right hom-ext perpendicular category of the underlying object $|T|\in \mathcal A$. Factorizations of a morphism $[T]$ are given by total orderings of the components of $T$. This is equivalent to a "signed exceptional sequence." For an algebra of finite representation type, the geometric realization of the cluster morphism category is an Eilenberg-MacLane space with fundamental group equal to the "picture group" introduced by the authors in [IOTW4].

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