Affine Insertion and Pieri Rules for the Affine Grassmannian

The authors study combinatorial aspects of the Schubert calculus of the affine Grassmannian ${ m Gr}$ associated with $SL(n,mathbb{C})$.Their main results are: Pieri rules for the Schubert bases of $H^*({ m Gr})$ and $H_*({ m Gr})$, which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. A new combinatorial definition for $k$-Schur functions, which represent the Schubert basis of $H_*({ m Gr})$. A combinatorial interpretation of the pairing $H^*({ m Gr}) imes H_*({ m Gr}) ightarrowmathbb Z$ induced by the cap product.