Geometric mitosis

TL;DR

This paper introduces a convex geometric algorithm for representing Schubert cycles in flag varieties, connecting combinatorics of polytopes with mitosis on pipe dreams, and extends to symplectic cases.

Contribution

It presents a novel convex geometric algorithm that realizes Schubert cycles via polytopes and extends combinatorial mitosis to symplectic flag varieties.

Findings

Algorithm coincides with mitosis on pipe dreams for GL_n.

Provides a new combinatorial rule for Sp_4 and extends to Sp_{2n}.

Connects geometric and combinatorial approaches in flag varieties.

Abstract

We describe an elementary convex geometric algorithm for realizing Schubert cycles in complete flag varieties by unions of faces of polytopes. For GL_n and Gelfand--Zetlin polytopes, combinatorics of this algorithm coincides with that of the mitosis on pipe dreams introduced by Knutson and Miller. For Sp_4 and a Newton--Okounkov polytope of the symplectic flag variety, the algorithm yields a new combinatorial rule that extends to Sp_{2n}.