2023/08/10 by Buch, Anders S., Chaput, Pierre-Emmanuel, Perrin, Nicolas · 2 citations
#14E08 (Secondary) #14M15 #14N15 #14N35 (Primary) 19E08 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2308.05307
We prove a collection of formulas for products of Schubert classes in the quantum K-theory ring QK(X) of a cominuscule flag variety X. This includes a K-theory version of the Seidel representation, stating that the quantum product of a Seidel class with an arbitrary Schubert class is equal to a single Schubert class times a power of the deformation parameter q. We also prove new Pieri formulas for the quantum K-theory of maximal orthogonal Grassmannians and Lagrangian Grassmannians, and give a new proof of the known Pieri formula for the quantum K-theory of Grassmannians of type A. Our formulas have simple statements in terms of quantum shapes that represent the natural basis elements qd[\mathcal OXu] of QK(X). Along the way we give a simple formula for K-theoretic Gromov-Witten invariants of Pieri type for Lagrangian Grassmannians, and prove a rationality result for the points in a Richardson variety in a symplectic Grassmannian that are perpendicular to a point in projective space.