Tropical critical points of the superpotential of a flag variety

Judd, Jamie

doi:10.1016/j.jalgebra.2017.11.019

Cited by 4 publications

(18 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However the proofs are very different and the representation theory arguments of [Kav15] do not extend to our setting. We note also a recent paper of Judd [Jud18], which adds detail to this picture in the case of SL n B.…”

mentioning

confidence: 68%

Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians

Rietsch¹,

Williams²

2019

Duke Math. J.

114

View full text Add to dashboard Cite

In this article we use cluster structures and mirror symmetry to explicitly describe a natural class of Newton-Okounkov bodies for Grassmannians. We consider the Grassmannian X = Gr n−k (C n ), as well as the mirror dual Landau-Ginzburg model (X ○ , W ∶X ○ → C), whereX ○ is the complement of a particular anti-canonical divisor in a Langlands dual GrassmannianX = Gr k ((C n ) * ), and the superpotential W has a simple expression in terms of Plücker coordinates [MR13]. Grassmannians simultaneously have the structure of an A-cluster variety and an X -cluster variety [Sco06, Pos]; roughly speaking, a cluster variety is obtained by gluing together a collection of tori along birational maps [FZ02, FG06]. Given a plabic graph or, more generally, a cluster seed G, we consider two associated coordinate systems: a network orare the open positroid varieties in X andX, respectively. To each X -cluster chart Φ G and ample 'boundary divisor' D in X ∖ X ○ , we associate a Newton-Okounkov body ∆ G (D) in R k(n−k) , which is defined as the convex hull of rational points; these points are obtained from the multi-degrees of leading terms of the Laurent polynomials Φ * G (f ) for f on X with poles bounded by some multiple of D. On the other hand using the A-cluster chart Φ ∨ G on the mirror side, we obtain a set of rational polytopes -described in terms of inequalities -by writing the superpotential W as a Laurent polynomial in the A-cluster coordinates, and then "tropicalising". Our first main result is that the Newton-Okounkov bodies ∆ G (D) and the polytopes obtained by tropicalisation on the mirror side coincide. As an application, we construct degenerations of the Grassmannian to normal toric varieties corresponding to (dilates of) these Newton-Okounkov bodies. Our second main result is an explicit combinatorial formula in terms of Young diagrams, for the lattice points of the Newton-Okounkov bodies, in the case that the cluster seed G corresponds to a plabic graph. This formula has an interpretation in terms of the quantum Schubert calculus of Grassmannians [FW04].

show abstract

mentioning

confidence: 68%

Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians

Rietsch¹,

Williams²

2019

Duke Math. J.

114

View full text Add to dashboard Cite

show abstract

“…Thus our main result, Theorem 1.1, can be viewed as a simultaneous generalisation of [Jud18] and [Gal97]. We also note that Galkin's theorem is in fact an ingredient that is used in our construction of the leading term coefficient of p crit .…”

mentioning

confidence: 76%

“…1.3.1. The first application of our result is the analogue for toric varieties of the result for flag varieties in [Jud18]. Namely if X Σ is a complete toric variety with a torus-invariant Weil divisor D, then we associate a positive, complete Laurent polynomial to X Σ and with it its tropical critical point, which, if D is ample, lies in the moment polytope P D .…”

mentioning

confidence: 85%

“…The interpretations in this section, and particularly Corollary 6.3, are a toric variety version of the first author's work in the setting of the flag variety GL n /B. In that case the tropical critical point of the mirror Landau-Ginzburg model, with quantum parameters q i = t λ,ω ∨ i , was computed directly and it picks out a special point in the Gelfand-Zetlin moment polytope of the representation V (λ), see [Jud18].…”

Section: Applications To Toric Varietiesmentioning

confidence: 99%

“…An answer to the question of Ginzburg is given in a precursor [Jud18] to this paper and involves computing the tropical critical point of Berenstein and Kazhdan's 'potential function' W λ BK from their theory of geometric crystals [BK07]. Namely [Jud18] proves the uniqueness and existence of a positive critical point in the special case of W λ BK , and shows that the special basis element in the 2ρ-representation is indexed by this tropical critical point for λ = 2ρ, which is a lattice point in a Gelfand-Zetlin polytope. We note that the Berenstein-Kazhdan potential function agrees with the superpotential mirror dual to the flag variety given in [Rie08].…”

mentioning

confidence: 99%

See 2 more Smart Citations

The tropical critical point and mirror symmetry

Judd,

Rietsch

2019

Preprint

Self Cite

View full text Add to dashboard Cite

Call a Laurent polynomial W 'complete' if its Newton polytope is full-dimensional with zero in its interior. We show that if W is any complete Laurent polynomial with coefficients in the positive part of the field K of generalised Puiseux series, then W has a unique positive critical point p crit . Here a generalised Puiseux series is called 'positive' if the coefficient of its leading term is in R >0 . Using the valuation on K we obtain from p crit ∈ K r >0 a canonically associated 'tropical critical point' d crit ∈ R r for which we give a finite recursive construction.One application, using results of [FOOO12], is that any symplectic toric manifold X ∆ with rational moment polytope has a canonically associated nondisplaceable Lagrangian torus, which can be explicitly constructed as a moment map fiber. This Lagrangian torus in X ∆ generalises the Clifford torus inside CP r .In the context of a complete toric variety X Σ our results construct a canonical point in the moment polytope of any ample line bundle O(D). If this point is a lattice point, we obtain a distinguished toric divisor in the divisor class [D]. This distinguished divisor generalises the standard toric boundary divisor to divisor classes other than the anti-canonical class.We also show that the positive critical point is preserved by mutation and thus is well-defined in the setting of cluster varieties. Finally, we show if the field K is replaced by the ordinary field of Puiseaux series K, then the positive critical point lies in K r >0 and d crit ∈ Q r . And in the other direction, we show that our theorem carries over to the case where the exponents v i of W are not integral but in R r , even though W is then no longer Laurent.

show abstract

Ideal polytopes for representations of GLn(C)

Lüdenbach

2024

Journal of Algebra

View full text Add to dashboard Cite

Tropical critical points of the superpotential of a flag variety

Cited by 4 publications

References 9 publications

Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians

Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians

The tropical critical point and mirror symmetry

Ideal polytopes for representations of GLn(C)

Contact Info

Product

Resources

About