Existence of symmetric maximal noncrossing collections of k-element sets

Pasquali, Andrea; Thörnblad, Erik; Zimmermann, Jakob

doi:10.1007/s10801-019-00893-8

Cited by 2 publications

(2 citation statements)

References 13 publications

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“…Theorem 6.7. [PTZ18] There exists an (a, n)-Postnikov diagram which is invariant under rotation by 2πa n if and only if a is congruent to -1, 0 or 1 modulo n/ GCD(n, a). In particular there are infinitely many self-injective planar QPs with Nakayama automorphism of order d, for any choice of d.…”

Section: Planar Rotation-invariant Qpsmentioning

confidence: 99%

Skew group algebras of Jacobian algebras

Giovannini

Pasquali

2019

Journal of Algebra

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For a quiver with potential (Q, W ) with an action of a finite cyclic group G, we study the skew group algebra ΛG of the Jacobian algebra Λ = P(Q, W ). By a result of Reiten and Riedtmann, the quiver Q G of a basic algebra η(ΛG)η Morita equivalent to ΛG is known. Under some assumptions on the action of G, we explicitly construct a potential W G on Q G such that η(ΛG)η ∼ = P(Q G , W G ). The original quiver with potential can then be recovered by the skew group algebra construction with a natural action of the dual group of G. If Λ is self-injective, then ΛG is as well, and we investigate this case. Motivated by Herschend and Iyama's characterisation of 2-representation finite algebras, we study how cuts on (Q, W ) behave with respect to our construction.2010 Mathematics Subject Classification. 16G20, 16S35.

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Section: Planar Rotation-invariant Qpsmentioning

confidence: 99%

Skew group algebras of Jacobian algebras

Giovannini

Pasquali

2019

Journal of Algebra

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“…As a small example, one knows that clusters in Gr(2, 8) correspond to triangulations of an octagon, and it is easy to see that the octagon admits no ρ 2 -invariant triangulations, hence no 2-clusters. Necessary and sufficient conditions for the existence of k-clusters in Gr(k, n) were given in [35]. The conditions depend on the value of k mod p. On the other hand, we have the following.…”

Section: Tp Testsmentioning

confidence: 99%

Cyclic symmetry loci in Grasssmannians

Fraser

2020

Preprint

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The Grassmannian Gr(k, n) admits an action by a finite cyclic group via the cyclic shift map. We give a simple description of the points fixed by each element of this cyclic group, extending Karp's description of the points fixed by the cyclic shift itself. We give a cell decomposition of the set of totally nonnegative points in each cyclic symmetry locus and describe efficient total positivity tests, extending results of Postnikov to the cyclically symmetric setting. We describe a conjectural generalized cluster structure on cyclic symmetry loci provided the order of the orbifold point is sufficiently large. The generalized exchange relations we find should be a Higher Teichmüller analogue of the relations Chekhov and Shapiro used to study Teichmüller theory of orbifolds. n k −1 as a closed subvariety of projective space in the following way. A choice of ordered basis for X ∈ Gr(k, n) determines a k × n matrix M whose rows are the basis vectors. Then the Plücker embedding Gr(k, n) ↪ P n k −1 sends X ↦ (∆ I (X)) I∈ [n] k ∶= (∆ I (M)) I∈ [n] k .

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Existence of symmetric maximal noncrossing collections of k-element sets

Cited by 2 publications

References 13 publications

Skew group algebras of Jacobian algebras

Skew group algebras of Jacobian algebras

Cyclic symmetry loci in Grasssmannians

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