Computing toric degenerations of flag varieties

We compute toric degenerations arising from the tropicalization of the full flag varieties \(\mathop{\mathrm{Fl}}\nolimits _{4}\) and \(\mathop{\mathrm{Fl}}\nolimits _{5}\) embedded in a product of Grassmannians. For \(\mathop{\mathrm{Fl}}\nolimits _{4}\) and \(\mathop{\mathrm{Fl}}\nolimits _{5}\) 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.