arXiv subjects

Khrystyna Serhiyenko

Publications and source records attributed to Khrystyna Serhiyenko.

At least 19 recordsLinked to original sources

arXiv2026

A geometric model for syzygies over 2-Calabi-Yau tilted algebras

In this article, we consider the class of 2-Calabi-Yau tilted algebras that are defined by a quiver with potential whose dual graph is a tree. We call these algebras \emph{dimer tree algebras} because they can also be realized as quotients of dimer algebras on a disc. These algebras are wild in general. For every such algebra $B$, we construct a polygon $\mathcal{S}$ with a checkerboard pattern in its interior that gives rise to a category $\text{Diag}(\mathcal{S})$. The indecomposable objects of $\text{Diag}(\mathcal{S})$ are the 2-diagonals in $\mathcal{S}$, and its morphisms are given by certain pivoting moves between the 2-diagonals. We conjecture that the category $\text{Diag}(\mathcal{S})$ is equivalent to the stable syzygy category over the algebra $B$, such that the rotation of the polygon corresponds to the shift functor on the syzygies. In particular, the number of indecomposable syzygies is finite and the projective resolutions are periodic. We prove the conjecture in the special case where every chordless cycle in the quiver is of length three. As a consequence, we obtain an explicit description of the projective resolutions. Moreover, we show that the syzygy category is equivalent to the 2-cluster category of type $\mathbb{A}$, and we introduce a new derived invariant for the algebra $B$ that can be read off easily from the quiver.

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arXiv2026

A geometric model for syzygies over 2-Calabi-Yau tilted algebras II

In this article, we continue the study of a certain family of 2-Calabi-Yau tilted algebras, called dimer tree algebras. The terminology comes from the fact that these algebras can also be realized as quotients of dimer algebras on a disc. They are defined by a quiver with potential whose dual graph is a tree, and they are generally of wild representation type. Given such an algebra $B$, we construct a polygon $\mathcal{S}$ with a checkerboard pattern in its interior, that defines a category $\text{Diag}(\mathcal{S})$. The indecomposable objects of $\text{Diag}(\mathcal{S})$ are the 2-diagonals in $\mathcal{S}$, and its morphisms are certain pivoting moves between the 2-diagonals. We prove that the category $\text{Diag}(\mathcal{S})$ is equivalent to the stable syzygy category of the algebra $B$. This result was conjectured by the authors in an earlier paper, where it was proved in the special case where every chordless cycle is of length three. As a consequence, we conclude that the number of indecomposable syzygies is finite, and moreover the syzygy category is equivalent to the 2-cluster category of type $\mathbb{A}$. In addition, we obtain an explicit description of the projective resolutions, which are periodic. Finally, the number of vertices of the polygon $\mathcal{S}$ is a derived invariant and a singular invariant for dimer tree algebras, which can be easily computed form the quiver.

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arXiv2026

On Gorenstein algebras of finite Cohen-Macaulay type: dimer tree algebras and their skew group algebras

Dimer tree algebras are a class of non-commutative Gorenstein algebras of Gorenstein dimension 1. In previous work we showed that the stable category of Cohen-Macaulay modules of a dimer tree algebra $A$ is a 2-cluster category of Dynkin type $\mathbb{A}$. Here we show that, if $A$ has an admissible action by the group $G$ with two elements, then the stable Cohen-Macaulay category of the skew group algebra $AG$ is a 2-cluster category of Dynkin type $\mathbb{D}$. This result is reminiscent of and inspired by a result by Reiten and Riedtmann, who showed that for an admissible $G$-action on the path algebra of type $\mathbb{A}$ the resulting skew group algebra is of type $\mathbb{D}$. Moreover, we provide a geometric model of the syzygy category of $AG$ in terms of a punctured polygon $\mathcal{P}$ with a checkerboard pattern in its interior, such that the 2-arcs in $\mathcal{P}$ correspond to indecomposable syzygies in $AG$ and 2-pivots correspond to morphisms. In particular, the dimer tree algebras and their skew group algebras are Gorenstein algebras of finite Cohen-Macaulay type $\mathbb{A}$ and $\mathbb{D}$ respectively. We also provide examples of types $\mathbb{E}_6,\mathbb{E}_7,$ and $\mathbb{E}_8$.

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arXiv2026

On syzygy categories over Iwanaga-Gorenstein algebras: Reduction, minimality and finiteness

We study 2-Calabi-Yau tilted algebras which are non-commutative Iwanaga-Gorenstein algebras of Gorenstein dimension 1. In particular, we are interested in their syzygy categories or equivalently the stable categories of Cohen-Macauley modules $\underline{\text{CMP}}$. First we show that if an algebra $A$ is Iwanaga-Gorenstein of Gorenstein dimension 1 then its stable category is generated under extensions by its radical $\text{rad}\,A$. Next, for a 2-Calabi-Yau tilted algebra $A$ we provide an explicit relationship between the $\underline{\text{CMP}}$ category of $A$ and its quotient $A/Ae_iA$ by an ideal generated by an idempotent $e_i$. Consequently, we obtain various equivalent characterizations of when the $\underline{\text{CMP}}$ category remains the same after passing to the quotient. We also obtain applications to two classes of algebras that are CM finite, the dimer tree algebras and their skew group algebras.

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arXiv2026

Locally anti-blocking $\mathbf{g}$-polytopes for flow polytopes

Given an acyclic directed graph (DAG), the space of strength one flows is a lattice polytope called the flow polytope of the DAG. If the DAG admits an ample framing, then the flow polytope is Gorenstein and it linearly projects onto a reflexive polytope called the $\mathbf{g}$-polytope. We provide a combinatorial characterization of amply framed DAGs that have a locally anti-blocking $\mathbf{g}$-polytope, and we characterize the minimal faces of the $\mathbf{g}$-polytope containing a fixed pair of vertices. We prove in this case that the unimodular triangulation of the $\mathbf{g}$-polytope induced by the DKK triangulation of the flow polytope is a pulling triangulation, and we characterize the pulling orders that yield the DKK triangulation. To prove our results, we introduce and study coherence diagrams, a combinatorial model of coherence for amply framed DAGs with locally anti-blocking $\mathbf{g}$-polytopes. We conclude by indicating possible extensions of these results to the setting of $\mathbf{g}$-polytopes for gentle Nakayama algebras.

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arXiv2025

Classical tilting and $τ$-tilting theory via duplicated algebras

$τ$-tilting theory can be thought of as a generalization of the classical tilting theory which allows mutations at any indecomposable summand of a support $τ$-tilting pair. Indeed, for any algebra $Λ$ its tilting modules $\text{tilt}\,Λ$ form a subposet of the support $τ$-tilting poset $\text{s}τ-\text{tilt}\,Λ$. We show that conversely the $τ$-tilting theory of an algebra $Λ$ can be naturally identified with the classical tilting theory of its duplicated algebra $\barΛ$ by establishing a poset isomorphism $\text{s}τ-\text{tilt}\,Λ\cong \text{tilt}\,\barΛ$. As a result, $τ$-tilting theory may be considered to be a special case of tilting theory. This extends the results of Assem-Brüstle-Schiffler-Todorov in the case of hereditary algebras. We also show that the product $\text{s}τ-\text{tilt}\,Λ\times \text{s}τ-\text{tilt}\,Λ$ embeds into the support $τ$-tilting poset of its duplicated algebra $\text{s}τ-\text{tilt}\,\barΛ$ as a collection of Bongartz intervals. As an application we obtain a similar inclusion on the level of maximal green sequences.

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arXiv2025

$SL_k$-Tilings and Paths in $\mathbb{Z}^k$

An $SL_k$-tiling is a bi-infinite array of integers having all adjacent $k\times k$ minors equal to one and all adjacent $(k+1)\times (k+1)$ minors equal to zero. Introduced and studied by Bergeron and Reutenauer, $SL_k$-tilings generalize the notion of Conway-Coxeter frieze patterns in the case $k=2$. In a recent paper, Short showed a bijection between bi-infinite paths of reduced rationals in the Farey graph and $SL_2$-tilings. We extend this result to higher $k$ by constructing a bijection between $SL_k$-tilings and certain pairs of bi-infinite strips of vectors in $\mathbb{Z}^k$ called paths. The key ingredient in the proof is the connection to Plücker friezes and Grassmannian cluster algebras. As an application, we obtain results about periodicity, duality, and positivity for tilings.

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arXiv2025

Consistent Dimer Models on Surfaces with Boundary

A dimer model is a quiver with faces embedded in a surface. We define and investigate notions of consistency for dimer models on general surfaces with boundary which restrict to well-studied consistency conditions in the disk and torus case. We define weak consistency in terms of the associated dimer algebra and show that it is equivalent to the absence of bad configurations on the strand diagram. In the disk and torus case, weakly consistent models are nondegenerate, meaning that every arrow is contained in a perfect matching; this is not true for general surfaces. Strong consistency is defined to require weak consistency as well as nondegeneracy. We prove that the completed as well as the noncompleted dimer algebra of a strongly consistent dimer model are bimodule internally 3-Calabi-Yau with respect to their boundary idempotents. As a consequence, the Gorenstein-projective module category of the completed boundary algebra of suitable dimer models categorifies the cluster algebra given by their underlying quiver. We provide additional consequences of weak and strong consistency, including that one may reduce a strongly consistent dimer model by removing digons and that consistency behaves well under taking dimer submodels.

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arXiv2024

Boundary Algebras of Positroids

A dimer model is a quiver with faces embedded into a disk. A consistent dimer model gives rise to a strand diagram, and hence to a positroid. The Gorenstein-projective module category over the completed boundary algebra of a dimer model was shown by Pressland to categorify a cluster structure on the corresponding positroid variety. Outside of the Grassmannian case, boundary algebras of dimer models are not well understood. We give an explicit description of the boundary algebra of a consistent dimer model as a quiver with relations calculated only from the data of the decorated permutation or, equivalently, Grassmann necklace of its positroid.

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arXiv2024

Triangulations of Flow Polytopes, Ample Framings, and Gentle Algebras

The cone of nonnegative flows for a directed acyclic graph (DAG) is known to admit regular unimodular triangulations induced by framings of the DAG. These triangulations restrict to triangulations of the flow polytope for strength one flows, which are called DKK triangulations. For a special class of framings called ample framings, these triangulations of the flow cone project to a complete fan. We characterize the DAGs that admit ample framings, and we enumerate the number of ample framings for a fixed DAG. We establish a connection between maximal simplices in DKK triangulations and $τ$-tilting posets for certain gentle algebras, which allows us to impose a poset structure on the dual graph of any DKK triangulation for an amply framed DAG. Using this connection, we are able to prove that for full DAGs, i.e., those DAGs with inner vertices having in-degree and out-degree equal to two, the flow polytopes are Gorenstein and have unimodal Ehrhart $h^\ast$-polynomials.

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arXiv2024

Super Caldero--Chapoton map for type $A$

One can explicitly compute the generators of a surface cluster algebra either combinatorially, through dimer covers of snake graphs, or homologically, through the CC-map applied to indecomposable modules over the appropriate algebra. Recent work by Musiker, Ovenhouse and Zhang used Penner and Zeitlin's decorated super Teichm{ü}ller theory to define a super version of the cluster algebra of type $A$ and gave a combinatorial formula to compute the even generators. We extend this theory by giving a homological way of explicitly computing these generators by defining a super CC-map for type $A$.

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arXiv2023

$3$D Farey graph, lambda lengths and $SL_2$-tilings

We explore a three-dimensional counterpart of the Farey tessellation and its relations to Penner's lambda lengths and $SL_2$-tilings. In particular, we prove a three-dimensional version of Ptolemy relation, and generalise results of Ian Short to classify tame $SL_2$-tilings over Eisenstein integers in terms of pairs of paths in the 3D Farey graph.

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arXiv2022

Leclerc's conjecture on a cluster structure for type A Richardson varieties

Leclerc constructed a conjectural cluster structure on Richardson varieties in simply laced types using cluster categories. We show that in type A, his conjectural cluster structure is in fact a cluster structure. We do this by comparing Leclerc's construction with another cluster structure on type A Richardson varieties due to Ingermanson. Ingermanson's construction uses the combinatorics of wiring diagrams and the Deodhar stratification. Though the two cluster structures are defined very differently, we show that the quivers coincide and clusters are related by the twist map for Richardson varieties, recently defined by Galashin--Lam.

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arXiv2021

Triangulations, order polytopes, and generalized snake posets

This work regards the order polytopes arising from the class of generalized snake posets and their posets of meet-irreducible elements. Among generalized snake posets of the same rank, we characterize those whose order polytopes have minimal and maximal volume. We give a combinatorial characterization of the circuits in these order polytopes and then conclude that every triangulation is unimodular. For a generalized snake word, we count the number of flips for the canonical triangulation of these order polytopes. We determine that the flip graph of the order polytope of the poset whose lattice of filters comes from a ladder is the Cayley graph of a symmetric group. Lastly, we introduce an operation on triangulations called twists and prove that twists preserve regular triangulations.

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arXiv2020

Maximal green sequences for string algebras

Maximal green sequences are important objects in representation theory, cluster algebras, and string theory. The two fundamental questions about maximal green sequences are whether a given algebra admits such sequences and, if so, does it admit only finitely many. We study maximal green sequences in the case of string algebras and give sufficient conditions on the algebra that ensure an affirmative answer to these questions.

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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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arXiv2018

Combinatorics of cluster structures in Schubert varieties

We give an explicit combinatorial description of cluster structures in Schubert varieties of the Grassmannian in terms of (target labelings of) Postnikov's plabic graphs. This description is a natural generalization of the description given by (Scott 2006) for the Grassmannian and has been believed by experts essentially since (Scott 2006), though the statement was not formally written down until (Müller-Speyer 2016). To prove this conjecture we use a result of (Leclerc 2016), who used the module category of the preprojective algebra to prove that coordinate rings of many Richardson varieties in the complete flag variety admit cluster structures. We also adapt a construction of (Karpman 2016) to build cluster seeds associated to reduced expressions. Further, we explicitly describe cluster structures in skew Schubert varieties using plabic graphs whose boundary vertices need not be labeled in cyclic order.

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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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