Tubular cluster algebras II: Exponential growth

Barot, Michael; Geiß, Christof; Jasso, Gustavo

doi:10.1016/j.jpaa.2012.12.012

Cited by 7 publications

(37 citation statements)

References 21 publications

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“…Remark 8. 3), and with what we prove for the other finite mutation type Grassmannians Gr (3,9) and Gr (4,8) in Section 9. In the case that d = gcd(k, n) = 1, then no Artin generators are defined, and the conjecture says that G + (Gr(k, n)) is a finite cyclic group generated by the twist.…”

Section: 2supporting

confidence: 72%

“…Our results in Gr (3,9) and Gr (4,8) also have antecedents. Barot, Geiss, and Jasso [4], as well as Felikson, Shapiro, Thomas, and Tumarkin [12], gave Ping-Pong lemma arguments for a relation between these cluster modular groups and PSL 2 (Z).…”

Section: Prior Worksupporting

confidence: 66%

“…Quasi-automorphisms are closely related to other notions of symmetry in cluster theory (cf. [1,2,4,15]). Of these, we will focus on the cluster modular group.…”

Section: Symmetries Of Cluster Algebrasmentioning

confidence: 99%

“…Besides a few small examples, these cluster algebras have infinitely many clusters and cluster variables. We record here the Grassmannians of finite type, and of finite mutation type: (3,9)] and C[ Gr (4,8)] are of infinite type but finite mutation type.…”

Section: Grassmannian Cluster Algebrasmentioning

confidence: 99%

“…Section 8 summarizes what is known about the cluster modular group of Gr(k, n) and makes a conjecture describing them. Section 9 narrows our focus to Gr (3,9) and Gr (4,8). We review web combinatorics and state the Fomin-Pylyavskyy conjectures for Gr (3, n).…”

Section: Introductionmentioning

confidence: 99%

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Braid group symmetries of Grassmannian cluster algebras

Fraser

2020

Sel. Math. New Ser.

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Let Gr • (k, n) ⊂ Gr(k, n) denote the open positroid stratum in the Grassmannian. We define an action of the extended affine d-strand braid group on Gr • (k, n) by regular automorphisms, for d the greatest common divisor of k and n. The action is by quasi-automorphisms of the cluster structure on Gr • (k, n), determining a homomorphism from the extended affine braid group to the cluster modular group for Gr(k, n). We also define a quasi-isomorphism between the Grassmannian Gr(k, rk) and the Fock-Goncharov configuration space of 2r-tuples of affine flags for SL k . This identifies the cluster variables, clusters, and cluster modular groups, in these two cluster structures.Fomin and Pylyavskyy proposed a description of the cluster combinatorics for Gr(3, n) in terms of Kuperberg's basis of non-elliptic webs. As our main application, we prove many of their conjectures for Gr (3,9) and give a presentation for its cluster modular group. We establish similar results for Gr(4, 8). These results rely on the fact that both of these Grassmannians have finite mutation type.2010 Mathematics Subject Classification. 13F60.

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Section: 2supporting

confidence: 72%

Section: Prior Worksupporting

confidence: 66%

“…Quasi-automorphisms are closely related to other notions of symmetry in cluster theory (cf. [1,2,4,15]). Of these, we will focus on the cluster modular group.…”

Section: Symmetries Of Cluster Algebrasmentioning

confidence: 99%

Section: Grassmannian Cluster Algebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Braid group symmetries of Grassmannian cluster algebras

Fraser

2020

Sel. Math. New Ser.

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Tubular cluster algebras I: categorification

Barot

Geiß

2011

Math. Z.

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We present a categorification of four mutation finite cluster algebras by the cluster category of the category of coherent sheaves over a weighted projective line of tubular weight type. Each of these cluster algebras which we call tubular is associated to an elliptic root system. We show that via a cluster character the cluster variables are in bijection with the positive real Schur roots associated to the weighted projective line. In one of the four cases this is achieved by the approach to cluster algebras of Fomin-Shapiro-Thurston using a 2-sphere with 4 marked points whereas in the remaining cases it is done by the approach of Geiss-Leclerc-Schröer using preprojective algebras.

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τ2-Stable tilting complexes over weighted projective lines

Jasso

2015

Advances in Mathematics

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Abstract. Let X be a weighted projective line and coh X the associated categoy of coherent sheaves. We classify the tilting complexes T in D b (coh X) such that τ 2 T ∼ = T , where τ is the Auslander-Reiten translation in D b (coh X).As an application of this result, we classify the 2-representation-finite algebras which are derived-equivalent to a canonical algebra. This complements Iyama-Oppermann's classification of the iterated tilted 2-representation-finite algebras. By passing to 3-preprojective algebras, we obtain a classification of the selfinjective cluster-tilted algebras of canonical-type. This complements Ringel's classification of the selfinjective cluster-tilted algebras.

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Tubular cluster algebras II: Exponential growth

Cited by 7 publications

References 21 publications

Braid group symmetries of Grassmannian cluster algebras

Braid group symmetries of Grassmannian cluster algebras

Tubular cluster algebras I: categorification

τ2-Stable tilting complexes over weighted projective lines

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