# Schubert calculus on Newton-Okounkov polytopes

## TL;DR

This paper explores how Newton-Okounkov polytopes serve as convex geometric models for Schubert calculus, linking algebraic cycles to faces of polytopes and their intersections across types A, B, and C.

## Contribution

It introduces a general framework connecting Schubert calculus with Newton-Okounkov polytopes and surveys specific realizations in multiple Lie types.

## Key findings

- Faces of polytopes represent Schubert cycles
- Intersection of faces models cycle intersection
- Framework applies to types A, B, and C

## Abstract

A Newton-Okounkov polytope of a complete flag variety can be turned into a convex geometric model for Schubert calculus. Namely, we can represent Schubert cycles by linear combinations of faces of the polytope so that the intersection product of cycles corresponds to the set-theoretic intersection of faces (whenever the latter are transverse). We explain the general framework and survey particular realizations of this approach in types A, B and C.