Restricting Schubert classes to symplectic Grassmannians using self-dual puzzles

TL;DR

This paper introduces a puzzle-based formula for expanding Schubert classes from Grassmannians to symplectic Grassmannians, extending previous tableau and algorithmic methods, and incorporates equivariant cohomology using quantum integrable systems techniques.

Contribution

It provides a novel puzzle-based approach for Schubert class restrictions to symplectic Grassmannians, extending existing formulas and including equivariant cohomology.

Findings

Puzzle formula generalizes previous tableau and algorithmic methods.

Usual Grassmannian puzzle pieces suffice for certain Schubert calculus cases.

Incorporates techniques from quantum integrable systems.

Abstract

Given a Schubert class on $Gr(k,V)$ where $V$ is a symplectic vector space of dimension $2n$, we consider its restriction to the symplectic Grassmannian $SpGr(k,V)$ of isotropic subspaces. Pragacz gave tableau formulae for positively computing the expansion of these $H^*(Gr(k,V))$ classes into Schubert classes of the target when $k=n$, which corresponds to expanding Schur polynomials into $Q$-Schur polynomials. Coskun described an algorithm for their expansion when $k\leq n$. We give a puzzle-based formula for these expansions, while extending them to equivariant cohomology. We make use of a new observation that usual Grassmannian puzzle pieces are already enough to do some $2$-step Schubert calculus, and apply techniques from quantum integrable systems (``scattering diagrams'').