Polyhedral parametrizations of canonical bases &amp; cluster duality

Genz, Volker; Koshevoy, Gleb A.; Schumann, Bea

doi:10.1016/j.aim.2020.107178

Cited by 10 publications

(12 citation statements)

References 24 publications

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“…7.4]. In a different context, the composition of Kamnitzer's bijections inducing a map S * i (λ) → L i (λ) was computed in [GKS17,Prop. 8.2] and turns out to coincide with the map G λ i .…”

Section: A Bijection Between String and Lusztig Datamentioning

confidence: 99%

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Combinatorics of Canonical Bases Revisited: String Data in Type A

Genz

Koshevoy

Schumann

2021

Transformation Groups

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We give a formula for the crystal structure on the integer points of the string polytopes and the *-crystal structure on the integer points of the string cones of type A for arbitrary reduced words. As a byproduct, we obtain defining inequalities for Nakashima–Zelevinsky string polytopes. Furthermore, we give an explicit description of the Kashiwara *-involution on string data for a special choice of reduced word.

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Section: A Bijection Between String and Lusztig Datamentioning

confidence: 99%

“…Littelmann-Berenstein-Zelevinsky string polytopes have a vast amount of applications. They are generalizations of Gelfand-Tsetlin polytopes [Lit98], and appear as Newton-Okounkov bodies for flag varieties [FFL17], [K15] and in Gross-Hacking -Keel-Kontsevich's construction of canonical bases for cluster varieties [BF16], [GKS17].…”

Section: Introductionmentioning

confidence: 99%

Combinatorics of Canonical Bases Revisited: String Data in Type A

Genz

Koshevoy

Schumann

2021

Transformation Groups

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“…The cone C i consists of all t ∈ R N such that for any reduced word j the last coordinate of i j (t) is non-negative. This fact was used in [13] to define a function ς i ∈ C[x ± j | j ∈ {1, 2, . .…”

Section: Introductionmentioning

confidence: 99%

“…The proofs of Theorem 2 and Proposition 3 are making use of the following fact proven in [13]. The function ς i is the pullback of the potential function of [11] on the big reduced double Bruhat cell (see Proposition 5) by an isomorphism of tori.…”

Section: Introductionmentioning

confidence: 99%

Redundancy in string cone inequalities and multiplicities in potential functions on cluster varieties

Koshevoy

Schumann

2022

J Algebr Comb

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“…While [RW19] use Newton–Okounkov bodies for Grassmannians directly, the construction given in [GHKK18] is more general and is expected to recover toric degenerations of many representation theoretic objects. For example, the case of flag varieties is addressed in [Mag20, BF19, GKS16, GKS20].…”

Section: Introductionmentioning

confidence: 99%

Toric degenerations of cluster varieties and cluster duality

Bossinger¹,

Frías-Medina²,

Magee³

et al. 2020

Compositio Math.

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We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: an $\mathcal {X}$-cluster variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed $\mathcal {X}$-cluster variety $\widehat {\mathcal {X} }$ to the toric variety associated to its g-fan. Moreover, we show that the fibers of this family are stratified in a natural way, with strata the specially completed $\mathcal {X}$-varieties encoded by $\operatorname {Star}(\tau )$ for each cone $\tau$ of the $\mathbf {g}$-fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to $\mathcal {A}_{\mathrm {prin}}$ of Gross, Hacking, Keel and Kontsevich [Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497–608], and the fibers cluster dual to $\mathcal {A} _t$. Finally, we give two applications. First, we use our construction to identify the toric degeneration of Grassmannians from Rietsch and Williams [Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J. 168 (2019), 3437–3527] with the Gross–Hacking–Keel–Kontsevich degeneration in the case of $\operatorname {Gr}_2(\mathbb {C} ^{5})$. Next, we use it to link cluster duality to Batyrev–Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.

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Polyhedral parametrizations of canonical bases & cluster duality

Cited by 10 publications

References 24 publications

Combinatorics of Canonical Bases Revisited: String Data in Type A

Combinatorics of Canonical Bases Revisited: String Data in Type A

Redundancy in string cone inequalities and multiplicities in potential functions on cluster varieties

Toric degenerations of cluster varieties and cluster duality

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