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. They obtain an explicit combinatorial description of the expansion of an ungraded \(k\)-Schur function into \(k+1\)-Schur functions. As a corollary, they 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