# Folding procedure for Newton-Okounkov polytopes of Schubert varieties

## TL;DR

This paper introduces a folding procedure for Newton-Okounkov polytopes of Schubert varieties, revealing connections between different types and providing new insights into crystal bases in representation theory.

## Contribution

It presents a novel folding method for Newton-Okounkov polytopes of Schubert varieties, linking polytopes across types and offering a new interpretation of Kashiwara's crystal basis similarity.

## Key findings

- Folding procedure relates Newton-Okounkov polytopes of different types.
- Provides a new interpretation of Kashiwara's crystal basis similarity.
- Connects geometric polytopes with representation theory concepts.

## Abstract

The theory of Newton-Okounkov polytopes is a generalization of that of Newton polytopes for toric varieties, and it gives a systematic method of constructing toric degenerations of a projective variety. In the case of Schubert varieties, their Newton-Okounkov polytopes are deeply connected with representation theory. Indeed, Littelmann's string polytopes and Nakashima-Zelevinsky's polyhedral realizations are obtained as Newton-Okounkov polytopes of Schubert varieties. In this paper, we apply the folding procedure to a Newton-Okounkov polytope of a Schubert variety, which relates Newton-Okounkov polytopes of Schubert varieties of different types. As an application of this result, we obtain a new interpretation of Kashiwara's similarity of crystal bases.