arXiv subjects

Matthew Pressland

Publications and source records attributed to Matthew Pressland.

15 recordsLinked to original sources

arXiv2024

Cluster structures via representation theory: cluster ensembles, tropical duality, cluster characters and quantisation

We develop a general theory of cluster categories, applying to a 2-Calabi-Yau extriangulated category $\mathcal{C}$ and cluster-tilting subcategory $\mathcal{T}$ satisfying only mild finiteness conditions. We show that the structure theory of $\mathcal{C}$ and the representation theory of $\mathcal{T}$ give rise to the rich combinatorial structures of seed data and cluster ensembles, via Grothendieck groups and homological algebra. We demonstrate that there is a natural dictionary relating cluster-tilting subcategories and their tilting theory to A-side tropical cluster combinatorics and, dually, relating modules over $\underline{\mathcal{T}}$ to the X-side; here $\underline{\mathcal{T}}$ is the image of $\mathcal{T}$ in the triangulated stable category of $\mathcal{C}$. Moreover, the exchange matrix associated to $\mathcal{T}$ arises from a natural map $p_{\mathcal{T}}\colon\mathrm{K}_0(\operatorname{mod}\underline{\mathcal{T}})\to\mathrm{K}_0(\mathcal{T})$ closely related to taking projective resolutions. Via our approach, we categorify many key identities involving mutation, g-vectors and c-vectors, including in infinite rank cases and in the presence of loops and 2-cycles. We are also able to define A- and X-cluster characters, which yield A- and X-cluster variables when there are no loops or 2-cycles, and which enable representation-theoretic proofs of cluster-theoretical statements. Continuing with the same categorical philosophy, we give a definition of a quantum cluster category, as a cluster category together with the choice of a map closely related to the adjoint of $p_{\mathcal{T}}$. Our framework enables us to show that any Hom-finite exact cluster category admits a canonical quantum structure, generalising results of Gei{\ss}--Leclerc--Schr\"oer.

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arXiv2024

Perfect matching modules, dimer partition functions and cluster characters

Cluster algebra structures for Grassmannians and their (open) positroid strata are controlled by a Postnikov diagram D or, equivalently, a dimer model on the disc, as encoded by either a bipartite graph or the dual quiver (with faces). The associated dimer algebra A, determined directly by the quiver with a certain potential, can also be realised as the endomorphism algebra of a cluster-tilting object in an associated Frobenius cluster category. In this paper, we introduce a class of A-modules corresponding to perfect matchings of the dimer model of D and show that, when D is connected, the indecomposable projective A-modules are in this class. Surprisingly, this allows us to deduce that the cluster category associated to D embeds into the cluster category for the appropriate Grassmannian. We show that the indecomposable projectives correspond to certain matchings which have appeared previously in work of Muller-Speyer. This allows us to identify the cluster-tilting object associated to D, by showing that it is determined by one of the standard labelling rules constructing a cluster of Plücker coordinates from D. By computing a projective resolution of every perfect matching module, we show that Marsh-Scott's formula for twisted Plücker coordinates, expressed as a dimer partition function, is a special case of the general cluster character formula, and thus observe that the Marsh-Scott twist can be categorified by a particular syzygy operation in the Grassmannian cluster category.

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arXiv2024

From frieze patterns to cluster categories

Motivated by Conway and Coxeter's combinatorial results concerning frieze patterns, we sketch an introduction to the theory of cluster algebras and cluster categories for acyclic quivers. The goal is to show how these more abstract theories provide a conceptual explanation for phenomena concerning friezes, principally integrality and periodicity.

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arXiv2024

Extriangulated ideal quotients, with applications to cluster theory and gentle algebras

We extend results of Brüstle-Yang on ideal quotients of 2-term subcategories of perfect derived categories of non-positive dg algebras to a relative setting. We find a new interpretation of such quotients: they appear as prototypical examples of a new construction of quotients of extriangulated categories by ideals generated by morphisms from injectives to projectives. We apply our results to Frobenius exact cluster categories and Higgs categories with suitable relative extriangulated structures, and to categories of walks related to gentle algebras. In all three cases, the extriangulated structures are well-behaved (they are 0-Auslander) and their quotients are equivalent to homotopy categories of two-term complexes of projectives over suitable finite-dimensional algebras.

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arXiv2024

Cluster categories for completed infinity-gons I: Categorifying triangulations

Paquette and Y{\i}ld{\i}r{\i}m recently introduced triangulated categories of arcs in completed infinity-gons, which are discs with an infinite closed set of marked points on their boundary. These categories have many features in common with the cluster categories associated to discs with different sets of marked points. In particular, they have (weak) cluster-tilting subcategories, which Paquette and Y{\i}ld{\i}r{\i}m show are in bijection with very special triangulations of the disc. This is in contrast to Igusa and Todorov's earlier work in the uncompleted case, in which every triangulation corresponds to a weak cluster-tilting subcategory. In this paper, we replace the triangulated structure of Paquette and Y{\i}ld{\i}r{\i}m's category by an extriangulated substructure and prove that, with this structure, the weak cluster-tilting subcategories are once again in bijection with triangulations. We further show that functorial finiteness of a weak cluster-tilting subcategory is equivalent to a very mild condition on the triangulation, which also appears in \c{C}anak\c{c}{\i} and Felikson's study of infinite rank cluster algebras from Teichm\"uller theory. By comparison with the combinatorics of triangulations, we are also able to characterise when weak cluster-tilting subcategories can be mutated in this new extriangulated category.

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arXiv2023

Reduction of Frobenius extriangulated categories

We describe a reduction technique for stably 2-Calabi--Yau Frobenius extriangulated categories $\mathcal{F}$ with respect to a functorially finite rigid subcategory $\mathcal{X}$. The reduction of such a category is another category $\mathcal{X}^{\perp_1}\subseteq\mathcal{F}$ of the same kind, whose cluster-tilting subcategories are those cluster-tilting subcategories $\mathcal{T}\subseteq\mathcal{F}$ such that $\mathcal{X}\subseteq\mathcal{T}$. This reduction operation generalises Iyama--Yoshino's reduction for 2-Calabi--Yau triangulated categories, which is recovered by passing to stable categories. Moreover, for a certain class of categories $\mathcal{F}$ and rigid objects $M$, we show that the relationship between $\mathcal{F}$ and $M^{\perp_1}$ may also be expressed in terms of internally Calabi--Yau algebras, in the sense of the third author. As an application, we give a conceptual proof of a result on frieze patterns originally obtained by the first author with Baur, Gratz, Serhiyenko, and Todorov.

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arXiv2023

Quasi-coincidence of cluster structures on positroid varieties

By work of a number of authors, beginning with Scott and culminating with Galashin and Lam, the coordinate rings of positroid varieties in the Grassmannian carry cluster algebra structures. In fact, they typically carry many such structures, the two best understood being the source-labelled and target-labelled structures, referring to how the initial cluster is computed from a Postnikov diagram or plabic graph. In this article, we show that these two cluster algebra structures quasi-coincide, meaning in particular that a cluster variable in one structure may be expressed in the other structure as the product of a cluster variable and a Laurent monomial in the frozen variables. This resolves a conjecture attributed to Muller and Speyer from 2017. The proof depends critically on categorification: of the relevant cluster algebra structures by the author, of perfect matchings and twists by the author with \c{C}anak\c{c}{\i} and King, and of quasi-equivalences of cluster algebras by Fraser and Keller. By similar techniques, we also show that Muller and Speyer's left twist map is a quasi-cluster equivalence from the target-labelled structure to the source-labelled structure.

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arXiv2022

Calabi-Yau properties of Postnikov diagrams

We show that the dimer algebra of a connected Postnikov diagram in the disc is bimodule internally 3-Calabi-Yau in the sense of the author's earlier work. As a consequence, we obtain an additive categorification of the cluster algebra associated to the diagram, which (after inverting frozen variables) is isomorphic to the homogeneous coordinate ring of a positroid variety in the Grassmannian by a recent result of Galashin and Lam. We show that our categorification can be realised as a full extension closed subcategory of Jensen-King-Su's Grassmannian cluster category, in a way compatible with their bijection between rank 1 modules and Plücker coordinates.

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arXiv2021

Mutation of frozen Jacobian algebras

We survey results on mutations of Jacobian algebras, while simultaneously extending them to the more general setup of frozen Jacobian algebras, which arise naturally from dimer models with boundary and in the context of the additive categorification of cluster algebras with frozen variables via Frobenius categories. As an application, we show that the mutation of cluster-tilting objects in various such categorifications, such as the Grassmannian cluster categories of Jensen-King-Su, is compatible with Fomin-Zelevinsky mutation of quivers. We also describe an extension of this combinatorial mutation rule allowing for arrows between frozen vertices, which the quivers arising from categorifications and dimer models typically have.

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arXiv2021

Special tilting modules for algebras with positive dominant dimension

We study certain special tilting and cotilting modules for an algebra with positive dominant dimension, each of which is generated or cogenerated (and usually both) by projective-injectives. These modules have various interesting properties, for example that their endomorphism algebras always have global dimension at most that of the original algebra. We characterise minimal d-Auslander-Gorenstein algebras and d-Auslander algebras via the property that these special tilting and cotilting modules coincide. By the Morita-Tachikawa correspondence, any algebra of dominant dimension at least 2 may be expressed (essentially uniquely) as the endomorphism algebra of a generator-cogenerator for another algebra, and we also study our special tilting and cotilting modules from this point of view, via the theory of recollements and intermediate extension functors.

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arXiv2018

On quiver Grassmannians and orbit closures for gen-finite modules

We show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module, whose tilted algebra B is related to A by a recollement. Let M be a gen-finite A-module, meaning there are only finitely many indecomposable modules generated by M. Using the canonical tilts of endomorphism algebras of suitable cogenerators associated to M, and the resulting recollements, we construct desingularisations of the orbit closure and quiver Grassmannians of M, thus generalising all results from previous work of Crawley-Boevey and the second author in 2017. We provide dual versions of the key results, in order to also treat cogen-finite modules.

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arXiv2017

Graded Frobenius cluster categories

Recently the first author studied multi-gradings for generalised cluster categories, these being 2-Calabi-Yau triangulated categories with a choice of cluster-tilting object. The grading on the category corresponds to a grading on the cluster algebra without coefficients categorified by the cluster category and hence knowledge of one of these structures can help us study the other. In this work, we extend the above to certain Frobenius categories that categorify cluster algebras with coefficients. We interpret the grading K-theoretically and prove similar results to the triangulated case, in particular obtaining that degrees are additive on exact sequences. We show that the categories of Buan, Iyama, Reiten and Scott, some of which were used by Geiss, Leclerc and Schroer to categorify cells in partial flag varieties, and those of Jensen, King and Su, categorifying Grassmannians, are examples of graded Frobenius cluster categories.

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arXiv2017

A categorification of acyclic principal coefficient cluster algebras

In earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables by starting from the data of an internally Calabi-Yau algebra, which becomes the endomorphism algebra of a cluster-tilting object in the resulting category. In this paper, we construct appropriate internally Calabi-Yau algebras for cluster algebras with polarised principal coefficients (which differ from those with principal coefficients by the addition of more frozen variables), and obtain Frobenius categorifications in the acyclic case. Via partial stabilisation, we then define extriangulated categories, in the sense of Nakaoka and Palu, categorifying acyclic principal coefficient cluster algebras, for which Frobenius categorifications do not exist in general. Many of the intermediate results used to obtain these categorifications remain valid without the acyclicity assumption, as we will indicate, and are interesting in their own right. Most notably, we provide a Frobenius version of Van den Bergh's result that the Ginzburg dg-algebra of a quiver with potential is bimodule 3-Calabi-Yau.

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arXiv2017

Internally Calabi-Yau algebras and cluster-tilting objects

We describe what it means for an algebra to be internally d-Calabi-Yau with respect to an idempotent. This definition abstracts properties of endomorphism algebras of (d-1)-cluster-tilting objects in certain stably (d-1)-Calabi-Yau Frobenius categories, as observed by Keller-Reiten. We show that an internally d-Calabi-Yau algebra satisfying mild additional assumptions can be realised as the endomorphism algebra of a (d-1)-cluster-tilting object in a Frobenius category. Moreover, if the algebra satisfies a stronger 'bimodule' internally d-Calabi-Yau condition, this Frobenius category is stably (d-1)-Calabi-Yau. We pay special attention to frozen Jacobian algebras; in particular, we define a candidate bimodule resolution for such an algebra, and show that if this complex is indeed a resolution, then the frozen Jacobian algebra is bimodule internally 3-Calabi-Yau with respect to its frozen idempotent. These results suggest a new method for constructing Frobenius categories modelling cluster algebras with frozen variables, by first constructing a suitable candidate for the endomorphism algebra of a cluster-tilting object in such a category, analogous to Amiot's construction in the coefficient-free case.

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arXiv2016

Labelled seeds and the mutation group

We study the set S of labelled seeds of a cluster algebra of rank n inside a field F as a homogeneous space for the group M_n of (globally defined) mutations and relabellings. Regular equivalence relations on S are associated to subgroups W of Aut_{M_n}(S), and we thus obtain groupoids W \ S. We show that for two natural choices of equivalence relation, the corresponding groups W^c and W^+ act on F, and the groupoids W^c \ S and W^+ \ S on the model field K=Q(x_1,...,x_n). The groupoid W^+ \ S is equivalent to Fock-Goncharov's cluster modular groupoid. Moreover, W^c is isomorphic to the group of cluster automorphisms, and W^+ to the subgroup of direct cluster automorphisms, in the sense of Assem-Schiffler-Shramchenko. We also prove that, for mutation classes whose seeds have mutation finite quivers, the stabilizer of a labelled seed under M_n determines the quiver of the seed up to 'similarity', meaning up to taking opposites of some of the connected components. Consequently, the subgroup W^c is the entire automorphism group of S in these cases.

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