Computing toric degenerations of flag varieties

Abstract

We compute toric degenerations arising from the tropicalization of the full flag varieties Fl4 and Fl5 embedded in a product of Grassmannians. For Fl4 and Fl5 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.