Memoirs of the American Mathematical Society 2013; 101 pp; softcover Volume: 223 ISBN10: 082187294X ISBN13: 9780821872949 List Price: US$72 Individual Members: US$43.20 Institutional Members: US$57.60 Order Code: MEMO/223/1050
 The authors give a combinatorial expansion of a Schubert homology class in the affine Grassmannian \(\mathrm{Gr}_{\mathrm{SL}_k}\) into Schubert homology classes in \(\mathrm{Gr}_{\mathrm{SL}_{k+1}}\). This is achieved by studying the combinatorics of a new class of partitions called \(k\)shapes, which interpolates between \(k\)cores and \(k+1\)cores. The authors define a symmetric function for each \(k\)shape, and show that they expand positively in terms of dual \(k\)Schur functions. The authors obtain an explicit combinatorial description of the expansion of an ungraded \(k\)Schur function into \(k+1\)Schur functions. As a corollary, the authors give a formula for the Schur expansion of an ungraded \(k\)Schur function. Table of Contents  Introduction
 The poset of \(k\)shapes
 Equivalence of paths in the poset of \(k\)shapes
 Strips and tableaux for \(k\)shapes
 Pushout of strips and row moves
 Pushout of strips and column moves
 Pushout sequences
 Pushouts of equivalent paths are equivalent
 Pullbacks
 Appendix A. Tables of branching polynomials
 Bibliography
