Affine approach to quantum Schubert calculus

By Alexander Postnikov. 2002

138 citations recorded by the source corpus.

This paper presents a formula for products of Schubert classes in the quantum cohomology ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials for shapes that are naturally embedded in a torus. Then we show that the coefficients in the expansion of these toric Schur polynomials, in terms of the regular Schur polynomials, are exactly the 3-point Gromov-Witten invariants, which are the structure constants of the quantum cohomology ring. This construction implies three symmetries of the Gromov-Witten invariants of the Grassmannian with respect to the groups S3, (Z/nZ)2, and Z/2Z. The last symmetry is a certain curious duality of the quantum cohomology which inverts the quantum parameter q . Our construction gives a solution to a problem posed by Fulton and Woodward about the characterization of the powers of the quantum parameter q which occur with nonzero coefficients in the quantum product of two Schubert classes. The curious duality switches the smallest such power of q with the highest power. We also discuss the affine nil-Temperley-Lieb algebra that gives a model for the quantum cohomology.

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

View Paper: https://www.semanticscholar.org/paper/bdaac8a7a8197a4405a0e45740d4edb3b9072d16

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 K-THEORY OF GRASSMANNIANS (2008, unknown): “A G ] 1 3 Ju n 20 09”
k-Schur Functions and Affine Schubert Calculus (2013, methodology): “In [133], Postnikov used the affine nil-Temperley–Lieb algebra, which is a quotient of the nil-Coxeter algebra, to provide a model for the quantum cohomology.”
On the quantum product of Schubert classes (2001, unknown): “The (complex) codimension of σλ is |λ| = ∑ λi, the number of boxes in the Young diagram.”
QUANTUM COHOMOLOGY AND THE k-SCHUR BASIS (2005, background): “Details of a connection between the dual k-Schur functions and the cylindric Schur functions of [25] is also carried out in [17].”
Giambelli formulae for the equivariant quantum cohomology of the Grassmannian (2005, background): “For other (combinatorial) proofs of the quantum Giambelli formula see [4, 29].”
Affine Stanley symmetric functions (2005, methodology): “Separately, cylindric Schur functions were defined by Postnikov [27] (see also [8]).”
Commutator length of symplectomorphisms (2001, unknown): “Postnikov obtained the following result [41].”
Cylindric skew Schur functions (2004, background): “The main result of [15] serves as the starting point for our results of the next section.”