String cone and superpotential combinatorics for flag and Schubert varieties in type A

@article{Bossinger2019StringCA,
  title={String cone and superpotential combinatorics for flag and Schubert varieties in type A},
  author={Lara Bossinger and Ghislain Fourier},
  journal={J. Comb. Theory, Ser. A},
  year={2019},
  volume={167},
  pages={213-256}
}
We study the combinatorics of pseudoline arrangements and their relation to the geometry of flag and Schubert varieties. We associate to each pseudoline arrangement two polyhedral cones, defined in a dual manner. We prove that one of them is the weighted string cone by Littelmann and Berenstein-Zelevinsky. For the other we show how it arises in the framework of cluster varieties and mirror symmetry by Gross-Hacking-Keel-Kontsevich: for the flag variety the cone is the tropicalization of their… Expand
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References

SHOWING 1-10 OF 51 REFERENCES
Combinatorics of canonical bases revisited: Type A
We initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type $A$. Using geometric objects called RhombicExpand
Crystal bases and Newton–Okounkov bodies
Let G be a connected reductive algebraic group. We prove that the string parametrization of a crystal basis for a finite dimensional irreducible representation of G coincides with a natural valuationExpand
Canonical bases for cluster algebras
In [GHK11], Conjecture 0.6, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonicalExpand
Polyhedral parametrizations of canonical bases & cluster duality
We establish the relation of the potential function constructed by Gross-Hacking-Keel-Kontsevich's and Berenstein-Kazhdan's decoration function on the open double Bruhat cell in the base affine spaceExpand
Degenerations of flag and Schubert varieties to toric varieties
In this paper, we prove the degenerations of Schubert varieties in a minusculeG/P, as well as the class of Kempf varieties in the flag varietySL(n)/B, to (normal) toric varieties. As a consequence,Expand
Khovanskii Bases, Higher Rank Valuations, and Tropical Geometry
TLDR
The notion of a Khovanskii basis for $(A, \mathfrak{v})$ is introduced which provides a framework for far extending Gr\"obner theory on polynomial algebras to general finitely generated algeBRas and construct an associated compactification of $Spec(A)$. Expand
Donaldson-Thomas transformation of double bruhat cells in semisimple Lie groups
Double Bruhat cells $G^{u,v}$ were studied by Fomin and Zelevinsky. They provide important examples of cluster algebras and cluster Poisson varieties. Cluster varieties produce examples of 3dExpand
Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory
Generalizing the notion of Newton polytope, we define the Newton-Okounkov body, respectively, for semigroups of integral points, graded algebras, and linear series on varieties. We prove that anyExpand
Birational geometry of cluster algebras
We give a geometric interpretation of cluster varieties in terms of blowups of toric varieties. This enables us to provide, among other results, an elementary geometric proof of the LaurentExpand
Toric degenerations of Schubert varieties
LetG be a simply connected semisimple complex algebraic group. We prove that every Schubert variety ofG has a flat degeneration into a toric variety. This provides a generalization of results of [9],Expand
...
1
2
3
4
5
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