Computing toric degenerations of flag varieties

TL;DR

This paper computes and compares toric degenerations of full flag varieties from tropicalization, string polytopes, and FFLV polytopes, providing a general method for non-prime initial ideals.

Contribution

It introduces a procedure to find toric degenerations even when the tropical initial ideal is not prime, and compares different degeneration methods for flag varieties.

Findings

Toric degenerations from tropicalization of $ ext{Fl}_4$ and $ ext{Fl}_5$ are computed.

Comparison of degenerations from string polytopes, FFLV polytope, and tropicalization.

A general method for non-prime initial ideals in toric degenerations is presented.

Abstract

We compute toric degenerations arising from the tropicalization of the full flag varieties $\mathcal{F}\ell_4$ and $\mathcal{F}\ell_5$ embedded in a product of Grassmannians. For $\mathcal{F}\ell_4$ and $\mathcal{F}\ell_5$ we compare toric degenerations arising from string polytopes and the FFLV polytope with those obtained from the tropicalization of the flag varieties. We also present a general procedure to find toric degenerations in the cases where the initial ideal arising from a cone of the tropicalization of a variety is not prime.