Quantum cohomology of the Lagrangian Grassmannian

By Andrew Kresch, Harry Tamvakis. 2003

72 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9b88a607bc25468ad6deed54f0787f1a30727451

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Quantum Pieri rules for isotropic Grassmannians (2008, background): “ontained in LG(m,U⊕V) ⊂ IG. The claim now follows from Proposition 1.1. The following theorem generalizes [BKT, Thm. 2]. The vanishing statement in part (d) was proved for Lagrangian Grassmannians in [KT1]. Theorem 1.3. Let d≥ 0 and choose λ,µ,ν∈ P(k,n) such that |λ| + |µ| + |ν| = dim(IG)+d(n+1+k). ”
Quantum Cohomology of Minuscule Homogeneous Spaces (2006, background): “In the case of isotropic Grassmannians, Kresch and Tamvakis have proved a more general result ([ KT1 , p. 808] and [KT2, p. 499]) for the quantum product of any Schubert class with the point class.”
On the quantum product of Schubert classes (2001, background): “of complete flags ([21], [16], [18], [15]) and partial flags ([2], [27], [17], [15]), and recently for the Lagrangian Grassmannian [32].”
Quantum Cohomology of Grassmannians (2001, background): “For now we will start from the basic assumption that the quantum ring is a well defined associative ring, and aim towards proving generalizations of the Pieri and Giambelli formulas stated above to reveal its structure.”
Gromov-Witten invariants on Grassmannians (2003, methodology): “In types B , C , and D , our result is used to give new proofs of the structure theorems for the quantum cohomology of Lagrangian and orthogonal Grassman-nians (denoted LG and OG ) which were recently obtained in [KT1] and [KT2].”