Push–Pull Operators on Convex Polytopes

Abstract

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 $A$.