Computing toric degenerations of flag varieties

We compute toric degenerations arising from the tropicalization of the full flag varieties $\mathcal{F}{\mathrm{\ell }}_4$ and $\mathcal{F}{\mathrm{\ell }}_5$ embedded in a product of Grassmannians. For $\mathcal{F}{\mathrm{\ell }}_4$ and $\mathcal{F}{\mathrm{\ell }}_5$ 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.