Seidel and Pieri products in cominuscule quantum K-theory

TL;DR

This paper develops new formulas for products in the quantum K-theory ring of cominuscule flag varieties, including Seidel and Pieri formulas, with applications to Gromov-Witten invariants and Richardson varieties.

Contribution

It introduces novel Seidel and Pieri formulas in quantum K-theory for cominuscule varieties, expanding computational tools and theoretical understanding.

Findings

Proved a K-theory version of the Seidel representation.

Derived new Pieri formulas for orthogonal and Lagrangian Grassmannians.

Provided a simple formula for K-theoretic Gromov-Witten invariants of Pieri type.

Abstract

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 $q^d[{\mathcal O}_{X^u}]$ 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.