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The Poset of $$k$$-Shapes and Branching Rules for $$k$$-Schur Functions
Thomas Lam, University of Michigan, Ann Arbor, MI, Luc Lapointe, Universidad de Talca, Chile, Jennifer Morse, Drexel University, Philadelphia, PA, and Mark Shimozono, Virginia Polytechnic Institute and State University, Blacksburg, VA
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Memoirs of the American Mathematical Society
2013; 101 pp; softcover
Volume: 223
ISBN-10: 0-8218-7294-X
ISBN-13: 978-0-8218-7294-9
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.

• 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