QUANTUM COHOMOLOGY AND THE k-SCHUR BASIS

By Luc Lapointe, Jennifer Morse. 2005

67 citations recorded by the source corpus.

We prove that structure constants related to Hecke algebras at roots of unity are special cases of k-Littlewood-Richardson coefficients associated to a product of k-Schur functions. As a consequence, both the 3-point Gromov-Witten invariants appearing in the quantum cohomology of the Grassmannian, and the fusion coefficients for the WZW conformal field theories associated to su(l) are shown to be k-Littlewood-Richardson coefficients. From this, Mark Shimozono conjectured that the k-Schur functions form the Schubert basis for the homology of the loop Grassmannian, whereas k-Schur coproducts correspond to the integral cohomology of the loop Grassmannian. We introduce dual k-Schur functions defined on weights of k-tableaux that, given Shimozono's conjecture, form the Schubert basis for the cohomology of the loop Grassmannian. We derive several properties of these functions that extend those of skew Schur functions.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8adb4eeec98cfcf02cff119ca663d3ec6842c9a6

How researchers see this paper

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

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Quantum cohomology of G/P and homology of affine Grassmannian (2007, methodology): “We use dw,λ,Pu,v instead of c w,λ,P u,v since Woodward’s result is stated only for the nonequivariant coefficients.”
Schubert polynomials for the affine Grassmannian (2006, methodology): “They were earlier introduced in a different manner in [15], where they were called dual k-Schur functions.”
Affine Stanley symmetric functions (2005, background): “known connection between quantum cohomology and the Verlinde algebra, essentially implies that the main results of [Pos] and [LM05] are equivalent.”
k-Schur Functions and Affine Schubert Calculus (2013, methodology): “Lapointe, Lascoux, and Morse [92] conjectured that the Macdonald symmetric functions expand positively in terms of k-Schur functions.”
Affine Insertion and Pieri Rules for the Affine Grassmannian (2006, methodology): “Coincidence with RSK as n→ ∞ 70 References 71”
A k-tableau characterization of k-Schur functions (2005, background): “Consequently, the three point Gromov-Witten invariants are none other than “k-Littlewood-Richardson coefficients” occurring in s(k)µ s (k) ν = ∑ λ cλ,kµν s (k) λ .”