Newton–Okounkov Polytopes of Type $A$ Flag Varieties of Small Ranks Arising from Cluster Structures

A flag variety is a smooth projective homogeneous variety. In this paper, we study Newton–Okounkov polytopes of the flag variety $\mathrm {Fl}(\mathbb C^4)$ arising from its cluster structure. More precisely, we present defining inequalities of such Newton–Okounkov polytopes of $\mathrm {Fl}(\mathbb C^4)$. Moreover, we classify these polytopes, establishing their equivalence under unimodular transformations.