Combinatorics of semi-toric degenerations of Schubert varieties in type C

TL;DR

This paper extends combinatorial models for Schubert calculus to type C using semi-toric degenerations and pipe dreams with self-crossings, providing new tools for understanding symplectic Schubert varieties.

Contribution

It introduces a novel combinatorial model for type C Schubert calculus based on pipe dreams with self-crossings and shows how to construct these using skew mitosis operators.

Findings

Established a combinatorial model for type C Schubert calculus.

Proved the construction of the model via skew mitosis operators.

Extended previous results to symplectic Gelfand-Tsetlin polytopes.

Abstract

An approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Using the polytope ring of the Gelfand-Tsetlin polytopes, Kiritchenko-Smirnov-Timorin realized each Schubert class as a sum of reduced Kogan faces. The first named author introduced a generalization of reduced Kogan faces to symplectic Gelfand-Tsetlin polytopes using a semi-toric degeneration of a Schubert variety, and extended the result of Kiritchenko-Smirnov-Timorin to type C case. In this paper, we introduce a combinatorial model to this type C generalization using a kind of pipe dream with self-crossings. As an application, we prove that the type C generalization can be constructed by skew mitosis operators.