Quantum cohomology of the Lagrangian Grassmannian

TL;DR

This paper presents a detailed description of the quantum cohomology ring of the Lagrangian Grassmannian, including formulas and algorithms for quantum Schubert calculus, advancing understanding of its algebraic structure.

Contribution

It provides a presentation of QH^*(LG) and establishes its structure via (Q^~)-polynomials, along with quantum Pieri and Giambelli formulas and computational algorithms.

Findings

Quantum cohomology ring QH^*(LG) is described explicitly.

Quantum Schubert calculus formulas are formulated.

Algorithms for computing quantum structure constants are developed.

Abstract

Let V be a symplectic vector space and LG be the Lagrangian Grassmannian which parametrizes maximal isotropic subspaces in V. We give a presentation for the (small) quantum cohomology ring QH^*(LG) and show that its multiplicative structure is determined by the ring of (Q^~)-polynomials. We formulate a "quantum Schubert calculus" which includes quantum Pieri and Giambelli formulas, as well as algorithms for computing the structure constants appearing in the quantum product of Schubert classes.