Gromov-Witten invariants on Grassmannians

By Anders Skovsted Buch, Andrew Kresch, Harry Tamvakis. 2003

131 citations recorded by the source corpus.

We prove that any three-point genus zero Gromov-Witten invariant on a type A Grassmannian is equal to a classical intersection number on a two-step flag variety. We also give symplectic and orthogonal analogues of this result; in these cases the two-step flag variety is replaced by a sub-maximal isotropic Grassmannian. Our theorems are applied, in type A, to formulate a conjectural quantum Littlewood-Richardson rule, and in the other classical Lie types, to obtain new proofs of the main structure theorems for the quantum cohomology of Lagrangian and orthogonal Grassmannians.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3ad038e21dad7a8cfd5e1f2680152a014e40f1a8

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

Citations in the corpus

Quantum cohomology of orthogonal Grassmannians (2003, methodology): “n) 2N′(λ,µ)τ µr(n,n)q where both sums are over µ⊃ λwith |µ| = |λ| + ksuch that µ/λis a horizontal strip, and the second sum is restricted to those µwith two parts equal to n. In recent work with Buch [BKT], we give a more direct proof of the quantum Pieri rule for OG, and the corresponding rule for ”
Quantum Pieri rules for isotropic Grassmannians (2008, methodology): “µ,σνid to be the number of rational maps f: P1 → IG of degree dsuch that f(0) ∈ Xλ(E • ), f(1) ∈ Xµ(F • ), and f(∞) ∈ Xν(G • ), for given isotropic ﬂags E • , F • , and G • in general position. As in [BKT], we will show that these morphisms are maps to a Lagrangian Grassmannian LG(d,2d) contained in”
Quantum Groups and Quantum Cohomology (2012, background): “ieties. Grassmann varieties are among the oldest objects of study in algebraic geometry; in particular, their quantum cohomology has been described by many authors from many diﬀerent angles, see e.g. [4, 11, 12, 44, 57, 71, 95, 100]. The modest goal of this section is to help the reader align his fa”
A Littlewood–Richardson rule for two-step flag varieties (2009, methodology): “Since we will use Proposition 1 and Corollary 1 of [5], we summarize them in terms of the geometric point of view offered by Corollary 2.”
k-Schur Functions and Affine Schubert Calculus (2013, background): “Knutson formulated a conjecture for the subset of quantum Littlewood–Richardson coefficients as presented in [29] in terms of puzzles [74].”
Quantum Cohomology of Minuscule Homogeneous Spaces (2006, methodology): “• • • • • • • • • • • • • • • • • • • • • • • • QF(bv) 3.4 Computing degree one Gromov-Witten invariants In this subsection, we explain how to apply the technics of A. Buch, A. Kresch and H. Tamvakis [BKT] to calculate degree one Gromov-Witten invariants. Lemma 3.7 Let X(w) and X(v∗) be two Schubert”
A k-tableau characterization of k-Schur functions (2005, background): “n = Hµ[X;1,1]. Exciting mathematics has sprung from the search for combinatorial interpretations of the Gromov-Witten invariants and the Macdonald coeﬃcients, however the conjectures remain open. See [3, 4, 5, 13, 14, 22, 23, 25] and [9, 10] for examples of recent progress in these directions. 2. De”
QUANTUM K-THEORY OF GRASSMANNIANS (2008, methodology): “A result of Buch, Kresch, and Tamvakis [10] reduced the computation of the (3-point, genus 0) Gromov-Witten invariants of a Grassmann variety to a computation in the ordinary cohomology of certain two-step flag manifolds.”
Quantum Cohomology of Minuscule Homogeneous Spaces III Semi-Simplicity and Consequences (2007, unknown): “But then the Giambelli type formula (see [BKT1], Theorem 7) immediately implies that ¯ σ ( λ ) = 2 z ( λ ) σ ( ι ( λ )).”
Curve neighborhoods of Schubert varieties (2013, methodology): “The study of this integer was suggested by Zak [27] and plays a fundamental role in Chaput, Manivel, and Perrin’s generalization [9] of the ‘quantum equals classical’ principle from [6].”
Finiteness of cominuscule quantum K-theory (2010, background): “The class of cominuscule varieties consists of Grassmannians of type A, Lagrangian Grassmannians, maximal orthogonal Grassmannians, and quadric hypersurfaces.”
QUANTUM COHOMOLOGY AND THE k-SCHUR BASIS (2005, result): “Although there has been progress in certain cases [ 2 ], [3], [4], [14], [15], [27], [28], [30], a combinatorial interpretation for the 3-point Gromov-Witten invariants in complete generality remains an open problem.”
Quantum cohomology of G/P and homology of affine Grassmannian (2007, methodology): “It would be interesting to compare Corollary 10.17 with the work of Buch, Kresch and Tamvakis [4] who exhibit the Gromov-Witten invariants of (classical, orthogonal and Lagrangian) Grassmannians as classical Schubert structure constants.”