Memoirs of the American Mathematical Society 2010; 82 pp; softcover Volume: 208 ISBN10: 0821846582 ISBN13: 9780821846582 List Price: US$71 Individual Members: US$42.60 Institutional Members: US$56.80 Order Code: MEMO/208/977
 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.
Table of Contents  Schubert bases of \(\mathrm{Gr}\) and symmetric functions
 Strong tableaux
 Weak tableaux
 Affine insertion and affine Pieri
 The local rule \(\phi_{u,v}\)
 Reverse local rule
 Bijectivity
 Grassmannian elements, cores, and bounded partitions
 Strong and weak tableaux using cores
 Affine insertion in terms of cores
 Bibliography
