Push-pull operators on convex polytopes

TL;DR

This paper introduces convex geometric push-pull operators inspired by Schubert calculus, enabling inductive construction of Newton-Okounkov polytopes and Minkowski sums of specific polytopes in type A.

Contribution

It defines convex geometric analogs of push-pull operators and applies them to construct Newton-Okounkov polytopes and Minkowski sums in a new, inductive manner.

Findings

Constructed Newton-Okounkov polytopes of Bott-Samelson varieties.

Developed Minkowski sum of Feigin-Fourier-Littelmann-Vinberg polytopes.

Established applications to convex geometric representation theory.

Abstract

A classical result of Schubert calculus is an inductive description of Schubert cycles using divided difference (or push-pull) operators in Chow rings. We define convex geometric analogs of push-pull operators and describe their applications to the theory of Newton-Okounkov convex bodies. Convex geometric push-pull operators yield an inductive construction of Newton-Okounkov polytopes of Bott-Samelson varieties. In particular, we construct a Minkowski sum of Feigin-Fourier-Littelmann-Vinberg polytopes using convex geometric push-pull operators in type A.