Séminaire Lotharingien de Combinatoire, 82B. (2019), 12 pp.

Iva Halacheva, Allen Knutson, and Paul Zinn-Justin

Restricting Schubert classes to symplectic Grassmannians using self-dual puzzles

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. Coşkun described an algorithm for their expansion when k<=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").

Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

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