# Birational sequences and the tropical Grassmannian

## TL;DR

This paper introduces iterated sequences for Grassmannians, linking them to the tropical Grassmannian and toric degenerations, providing a recursive description of Newton--Okounkov polytopes and unifying various toric degenerations.

## Contribution

It presents a new class of birational sequences for Grassmannians that connect to the tropical Grassmannian and toric degenerations, offering a recursive method to describe Newton--Okounkov polytopes.

## Key findings

- Iterated sequences give rise to points in the tropical Grassmannian.
- Associated valuations induce toric degenerations for Gr(2,C^n).
- All toric degenerations from the tropical Grassmannian are recoverable via iterated sequences.

## 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.