A categorification of Grassmannian cluster algebras

Jensen, Bernt Tore; King, Alastair; Su, Xiuping

doi:10.1112/plms/pdw028

Cited by 57 publications

(146 citation statements)

References 23 publications

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“…Then F (or its stable part) is a cluster category. To prove this, [4] shows that (k, n)-diagrams give rise to elements of F: each alternating region in such a diagram D corresponds to a certain indecomposable module in F and the direct sum of these indecomposable modules over all the alternating region of D is a cluster-tilting object.…”

Section: A Module Category With Grassmannian Structurementioning

confidence: 99%

“…, n forming a cycle, connected by 2n arrows x i : i − 1 → i, y i : i → i + 1 and where R is the following set of relations: xy = yx (at every vertex) and x k = y n−k (at every vertex). By [4], these algebras give rise to additive categorifications of the cluster algebra structure of the Grassmannian: let F be the category of maximal Cohen-Macaulay-modules over B k,n . Then F (or its stable part) is a cluster category.…”

Section: A Module Category With Grassmannian Structurementioning

confidence: 99%

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Dimer models and Grassmannian cluster categories

Baur

2016

Proc Appl Math and Mech

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A dimer model can be defined as a quiver embedded into a surface in such a way that the complement is a disjoint union of disks with oriented boundaries. Such models can also be considered in the case of a surface with boundary. The Postnikov diagrams used by J. Scott to describe the cluster structure of the homogeneous coordinate ring of the Grassmannian give rise to dimer models on a disk in this sense. We associate a natural algebra to such a dimer model. This algebra is a modified version of the corresponding Jacobian algebra, taking the boundary into account. Taking the sum of the idempotents corresponding to boundary vertices, we obtain an idempotent subalgebra, which we call the boundary algebra. We show that it is independent of the choice of dimer model and coincides with an algebra that B. Jensen, A. King and X. Su have used to model the cluster structure of the homogeneous coordinate ring of the Grassmannian categorically. This reports on joint work with A. King (Bath) and R. Marsh (Leeds) and with D. Bogdanic (Graz).

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Section: A Module Category With Grassmannian Structurementioning

confidence: 99%

Section: A Module Category With Grassmannian Structurementioning

confidence: 99%

Dimer models and Grassmannian cluster categories

Baur

2016

Proc Appl Math and Mech

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“…In particular, CM(B) is closed under kernels of epimorphisms. Moreover [28,Cor. 3.7], B ∈ CM(B), and so Ω(mod B) ⊆ CM(B).…”

Section: Corollary 310 Let E Be a Frobenius M-cluster Category And Lmentioning

confidence: 99%

“…Example 3.12 Our second family of examples was introduced by Jensen-King-Su [28] to categorify the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian G n k of k-planes in C n . Each category in this family is of the form CM(B) for a Gorenstein order B (depending on positive integers 1…”

Section: Corollary 310 Let E Be a Frobenius M-cluster Category And Lmentioning

confidence: 99%

“…Thus the stable category E should be a cluster category-in particular it should be 2-Calabi-Yau. Such categorifications have been described for certain cluster algebras relating to partial flag varieties, for example by Geiss-Leclerc-Schröer [19], Jensen-King-Su [28] and Demonet-Iyama [14], and we will recall some of these constructions in Sect. 3.…”

Section: Introductionmentioning

confidence: 99%

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Internally Calabi–Yau algebras and cluster-tilting objects

Pressland

2017

Math. Z.

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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)-clustertilting 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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Conway–Coxeter Friezes and Mutation: A Survey

Baur

Faber

Gratz

et al. 2018

Association for Women in Mathematics Series

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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 clus-arXiv:1806.06441v1 [math.RA]

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A categorification of Grassmannian cluster algebras

Cited by 57 publications

References 23 publications

Dimer models and Grassmannian cluster categories

Dimer models and Grassmannian cluster categories

Internally Calabi–Yau algebras and cluster-tilting objects

Conway–Coxeter Friezes and Mutation: A Survey

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