Newton-Okounkov polytopes of type $A$ flag varieties of small ranks arising from cluster structures

Abstract

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