Quantum cohomology and the 𝑘-Schur basis

We prove that structure constants related to Hecke algebras at roots of unity are special cases of k k -Littlewood-Richardson coefficients associated to a product of k 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 s u ^ ( ℓ ) \widehat {su}(\ell ) are shown to be k k -Littlewood-Richardson coefficients. From this, Mark Shimozono conjectured that the k k -Schur functions form the Schubert basis for the homology of the loop Grassmannian, whereas k k -Schur coproducts correspond to the integral cohomology of the loop Grassmannian. We introduce dual k k -Schur functions defined on weights of k 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.