Abstract

We prove that structure constants related to Hecke algebras at roots of unity are special cases of ?-Littlewood-Richardson coefficients associated to a product of ?-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(?) are shown to be ?-Littlewood-Richardson coefficients. From this, Mark Shimozono conjectured that the ?-Schur functions form the Schubert basis for the homology of the loop Grassmannian, whereas ?-Schur coproducts correspond to the integral cohomology of the loop Grassmannian. We introduce dual ?-Schur functions defined on weights of ?-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. © 2007 American Mathematical Society.