Strong positivity for quantum theta bases of quantum cluster algebras

Davison, Ben; Mandel, Travis

doi:10.48550/arxiv.1910.12915

Cited by 11 publications

(29 citation statements)

References 57 publications

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“…He has since taken this further with Mandel. In [DM19] they show the strong positivity conjecture for quantum theta bases for skew-symmetric type quantum cluster algebras, meaning that the structure constants for decomposing products of quantum theta functions in this setting are also Laurent polynomials in the quantum parameter with positive integer coefficients. Here we give an example illustrating the failure of the quantum positivity conjecture for quantum theta functions outside of the skew-symmetric case.…”

Section: Non-positivity For Theta Functionsmentioning

confidence: 93%

“…The geometry and combinatorics of cluster varieties and their associated cluster algebras can be encoded using scattering diagrams, as formulated in [GHKK18]. In this section, we review the definition of scattering diagrams over skew-symmetric Lie algebras as in [KS14], [GHKK18], [CM19], [DM19]. We then use this machinery to discuss quantum scattering diagrams in Section 2.4.…”

Section: Scattering Diagramsmentioning

confidence: 99%

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Quantization of deformed cluster Poisson varieties

Cheung¹,

Frías-Medina²,

Magee³

2020

Preprint

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Fock and Goncharov described a quantization of cluster X -varieties (also known as cluster Poisson varieties) in [FG09]. Meanwhile, families of deformations of cluster X -varieties were introduced in [BFMN18]. In this paper we show that the two constructions are compatible-we extend the Fock-Goncharov quantization of X -varieties to the families of [BFMN18]. As a corollary, we obtain that these families and each of their fibers have Poisson structures. We relate this construction to the Berenstein-Zelevinsky quantization of A-varieties ([BZ05]). Finally, inspired by the counter-example to quantum positivity of the quantum greedy basis in [LLRZ14], we compute a counter-example to quantum positivity of the quantum theta basis.

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Section: Non-positivity For Theta Functionsmentioning

confidence: 93%

Section: Scattering Diagramsmentioning

confidence: 99%

Quantization of deformed cluster Poisson varieties

Cheung¹,

Frías-Medina²,

Magee³

2020

Preprint

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“…Remark 2.2.1. The formal completion k[L] can be shown to be the Cauchy completion with respect to an explicit topology on k[L], see [DM19]. It would be interesting to understand calculation in k[L] from a topological point of view, though we will not pursue this direction in this paper.…”

Section: Seeds As Basesmentioning

confidence: 99%

“…Then any Z ∈ LP(t) has a unique ≺ t -decomposition in terms of S (Definition 3.3.1). Notice that S is a topological basis in the sense of [DM19].…”

Section: Triangular Basesmentioning

confidence: 99%

Dual canonical bases and quantum cluster algebras

Fan¹

2020

Preprint

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Given any quantum cluster algebra arising from a quantum unipotent subgroup of symmetrizable Kac-Moody type, we verify the quantization conjecture in full generality that the quantum cluster monomials are contained in the dual canonical basis after rescaling. FAN QIN 6.5. Criteria for the existence of the common triangular bases 40 7. Prerequisite for quantum unipotent subgroup 43 7.1. Quantized enveloping algebras 43 7.2. Quantum unipotent subgroups 47 7.3. Dual canonical bases 50 7.4. Subalgebras and unipotent quantum minors 51 7.5. Localization 52 8. Cluster structures on quantum unipotent subgroup 53 8.1. CGL extension 53 8.2. Quantum cluster structure 55 8.3. Quantum cluster variables 59 9. Dual canonical bases are common triangular bases 61 9.1. Parametrization 62 9.2. Integral form 64 9.3. Localization and the initial triangular basis 66 9.4. Twist automorphism 67 9.5. Consequences 70 References 71 1 The conjecture was called the quantization conjecture by Kimura after the works [GLS11, GLS12] on (classical) unipotent subgroups. 2 The matrices P E,ε and P I uf F,ε are denoted by F ε and E ε in [Kel12, Section 5.6][BZ05, (3.2) (3.3)] respectively.

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“…More recently, the quantum CSDs were studied by Davison-Mandel [DM21], where the counterparts of many results presented here were given.…”

Section: Introduction To Part IIImentioning

confidence: 98%

Cluster Algebras and Scattering Diagrams, Part III. Cluster Scattering Diagrams

Наканиши¹

2021

Preprint

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This is a self-contained exposition of several fundamental properties of cluster scattering diagrams introduced and studied by Gross, Hacking, Keel, and Kontsevich. In particular, detailed proofs are presented for the construction, the mutation invariance, and the positivity of theta functions of cluster scattering diagrams. Throughout the text we highlight the fundamental roles of the dilogarithm elements and the pentagon relation in cluster scattering diagrams. * This is a preliminary draft of Part III of the forthcoming monograph "Cluster Algebras and Scattering Diagrams" by the author. Any comments are welcome.

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Strong positivity for quantum theta bases of quantum cluster algebras

Cited by 11 publications

References 57 publications

Quantization of deformed cluster Poisson varieties

Quantization of deformed cluster Poisson varieties

Dual canonical bases and quantum cluster algebras

Cluster Algebras and Scattering Diagrams, Part III. Cluster Scattering Diagrams

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