Schubert calculus from polyhedral parametrizations of Demazure crystals

TL;DR

This paper advances Schubert calculus by linking Demazure crystals with polyhedral parametrizations, enabling explicit combinatorial descriptions and applications to symplectic Gelfand-Tsetlin polytopes.

Contribution

It introduces explicit string parametrizations of opposite Demazure crystals and connects them with reduced Kogan faces via mitosis operators, extending Schubert calculus methods.

Findings

Explicit string parametrizations of Demazure crystals.

Relationship between Kogan faces and Demazure crystals.

Development of Schubert calculus on symplectic Gelfand-Tsetlin polytopes.

Abstract

One approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Through an identification of the cohomology ring of the type A full flag variety with the polytope ring of the Gelfand-Tsetlin polytopes, Kiritchenko-Smirnov-Timorin realized each Schubert class as a sum of reduced (dual) Kogan faces. In this paper, we explicitly describe string parametrizations of opposite Demazure crystals, which give a natural generalization of reduced dual Kogan faces. We also relate reduced Kogan faces with Demazure crystals using the theory of mitosis operators, and apply these observations to develop the theory of Schubert calculus on symplectic Gelfand-Tsetlin polytopes.