# Birational sequences and the tropical Grassmannian

@article{Bossinger2019BirationalSA,
title={Birational sequences and the tropical Grassmannian},
author={Lara Bossinger},
journal={arXiv: Representation Theory},
year={2019}
}
We introduce iterated sequences for Grassmannians, a new class of Fang-Fourier-Littelmanns' birational sequences and explain how they give rise to points in $\text{trop}(\text{Gr}(k,\mathbb C^n))$, Speyer-Sturmfels' tropical Grassmannian. For $\text{Gr}(2,\mathbb C^n)$ we show that the associated valuations induce toric degenerations. We describe recursively the vertices of the corresponding Newton--Okounkov polytopes, which are particular vertices of a hypercube and hence integral. We show… Expand
1 Citations

#### Figures and Tables from this paper

Full-Rank Valuations and Toric Initial Ideals
Let $V(I)$ be a polarized projective variety or a subvariety of a product of projective spaces and let $A$ be its (multi-)homogeneous coordinate ring. Given a full-rank valuation $\mathfrak v$ on $A$Expand

#### References

SHOWING 1-10 OF 25 REFERENCES
Essential bases and toric degenerations arising from birational sequences
• Mathematics
• 2015
We present a new approach to construct $T$-equivariant flat toric degenerations of flag varieties and spherical varieties, combining ideas coming from the theory of Newton-Okounkov bodies with ideasExpand
Khovanskii Bases, Higher Rank Valuations, and Tropical Geometry
• Mathematics, Computer Science
• SIAM J. Appl. Algebra Geom.
• 2019
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
Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians
• Mathematics
• 2019
We use cluster structures and mirror symmetry to explicitly describe a natural class of Newton-Okounkov bodies for Grassmannians. We consider the Grassmannian $X=Gr_{n-k}(\mathbb C^n)$, as well asExpand
Full-Rank Valuations and Toric Initial Ideals
Let $V(I)$ be a polarized projective variety or a subvariety of a product of projective spaces and let $A$ be its (multi-)homogeneous coordinate ring. Given a full-rank valuation $\mathfrak v$ on $A$Expand
PBW filtration and bases for irreducible modules in type An
• Mathematics
• 2010
We study the PBW filtration on the highest weight representations V(λ) of $$\mathfrak{s}{\mathfrak{l}_{n + 1}}$$. This filtration is induced by the standard degree filtration on $${\text{U}}\left(Expand Okounkov bodies and toric degenerations Let$$\varDelta $$be the Okounkov body of a divisor$$D$$on a projective variety$$X$$. We describe a geometric criterion for$$\varDelta  to be a lattice polytope, and show that in thisExpand
Transformation of Groups
In this work, we have proved a number of purely geometric statements by algebraic methods. Also we have proved Sylvester’s law of Nullity and Exercise: the nullity of the product BA never exceeds theExpand
Toric Degenerations of Gr(2, n) and Gr(3, 6) via Plabic Graphs
• Mathematics
• 2016
We establish an explicit bijection between the toric degenerations of the Grassmannian Gr(2, n) arising from maximal cones in tropical Grassmannians and the ones coming from plabic graphsExpand
Degenerations of flag and Schubert varieties to toric varieties
• Mathematics
• 1996
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
Toric degenerations of spherical varieties
• Mathematics
• 2004
Abstract.We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by mirror symmetry, weExpand