QUANTUM COHOMOLOGY AND THE k-SCHUR BASIS

@article{Lapointe2007QUANTUMCA,
  title={QUANTUM COHOMOLOGY AND THE k-SCHUR BASIS},
  author={Luc Lapointe and Jennifer Morse},
  journal={Transactions of the American Mathematical Society},
  year={2007},
  volume={360},
  pages={2021-2040}
}
  • L. Lapointe, J. Morse
  • Published 28 January 2005
  • Mathematics
  • Transactions of the American Mathematical Society
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… 
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References

SHOWING 1-10 OF 52 REFERENCES
Affine approach to quantum Schubert calculus
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
Schubert polynomials for the affine Grassmannian
Let G be a complex simply connected simple group and K a maximal compact subgroup. Let F = C((t)) denote the field of formal Laurent series and O = C[[t]] the ring of formal power series. The
The nil Hecke ring and cohomology of G/P for a Kac-Moody group G.
TLDR
A ring R is constructed, which is very simply and explicitly defined as a functor of W together with the W-module [unk] alone and such that all these four structures on H(*)(G/B) arise naturally from the ring R.
Gromov-Witten invariants on Grassmannians
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
Fermionic Formulas for Level-Restricted Generalized Kostka Polynomials and Coset Branching Functions
Abstract: Level-restricted paths play an important rôle in crystal theory. They correspond to certain highest weight vectors of modules of quantum affine algebras. We show that the recently
Order Ideals in Weak Subposets of Young’s Lattice and Associated Unimodality Conjectures
AbstractThe k-Young lattice Yk is a weak subposet of the Young lattice containing partitions whose first part is bounded by an integer k > 0. The Yk poset was introduced in connection with
Quantum Invariants of Knots and 3-Manifolds
This monograph, now in its second revised edition, provides a systematic treatment of topological quantum field theories in three dimensions, inspired by the discovery of the Jones polynomial of
...
...