Simple geometric mitosis

TL;DR

This paper introduces geometric operations on polytopes that unify and simplify the construction of mitosis procedures in Schubert calculus, applicable to Gelfand-Zetlin polytopes in types A and C.

Contribution

It presents a novel geometric framework that generalizes and simplifies existing mitosis constructions in Schubert calculus.

Findings

Provides a uniform geometric construction of Knutson-Miller mitosis in type A.

Simplifies Fujita mitosis in type C using geometric operations.

Establishes a connection between convex geometry and algebraic combinatorics.

Abstract

We construct simple geometric operations on faces of the Cayley sum of two polytopes. These operations can be thought of as convex geometric counterparts of divided difference operators in Schubert calculus. We show that these operations give a uniform construction of Knutson-Miller mitosis (in type A) and (simplified) Fujita mitosis (in type C) on Kogan faces of Gelfand-Zetlin polytopes.