Birational sequences and the tropical Grassmannian

Abstract

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 further that every toric degeneration of $\text{Gr}(2,{\mathbb{C}}^n)$ constructed using the tropical Grassmannian can be recovered by iterated sequences.

15 pages, 3 figures, 1 table, 1 algorithm. arXiv admin note: text overlap with arXiv:1806.02090

References in corpus (4)

• Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians
• Toric degenerations of spherical varieties
• Full-rank Valuations and Toric Initial Ideals
• Toric degenerations: a bridge between representation theory, tropical geometry and cluster algebras

Cited by in corpus (2)

• Full-rank Valuations and Toric Initial Ideals
• Polyhedral and Tropical Geometry of Flag Positroids