Affine Insertion and Pieri Rules for the Affine Grassmannian

The authors study combinatorial aspects of the Schubert calculus of the affine Grassmannian ${{\rm Gr}}$ associated with $SL(n,\mathbb{{C}})$.Their main results are: Pieri rules for the Schubert bases of $H^*({{\rm Gr}})$ and $H_*({{\rm Gr}})$, 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 $H_*({{\rm Gr}})$. A...
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The authors study combinatorial aspects of the Schubert calculus of the affine Grassmannian ${{\rm Gr}}$ associated with $SL(n,\mathbb{{C}})$.Their main results are: Pieri rules for the Schubert bases of $H^*({{\rm Gr}})$ and $H_*({{\rm Gr}})$, 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 $H_*({{\rm Gr}})$. A combinatorial interpretation of the pairing $H^*({{\rm Gr}})\times H_*({{\rm Gr}}) \rightarrow\mathbb Z$ induced by the cap product.