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

TL;DR

This paper investigates Newton-Okounkov polytopes of the flag variety of complex 4-space derived from its cluster structure, providing explicit inequalities and classifying their equivalence under unimodular transformations.

Contribution

It presents explicit defining inequalities for these polytopes and classifies their equivalence, advancing understanding of the geometric and combinatorial structure of flag varieties.

Findings

Explicit inequalities for Newton-Okounkov polytopes of $Fl(\mathbb{C}^4)$

Classification of polytopes up to unimodular transformations

Establishment of the polytopes' structural properties

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.