Restricting Schubert classes to symplectic Grassmannians using self-dual puzzles

2018/11/19 by Halacheva, Iva, Knutson, Allen, Zinn-Justin, Paul
#05E10 #14M15 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1811.07581

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≤ 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'').

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