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The poset of $k$-shapes and branching rules for $k$-Schur functions


About this Title

Thomas Lam, Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109, Luc Lapointe, Instituto de Matemática Y Física, Universidad de Talca, Casilla 747, Talca, Chile., Jennifer Morse, Department of Mathematics, Drexel University, Philadelphia, Pennsylvania 19104 and Mark Shimozono, Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061

Publication: Memoirs of the American Mathematical Society
Publication Year 2013: Volume 223, Number 1050
ISBNs: 978-0-8218-7294-9 (print); 978-0-8218-9874-1 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2012-00655-1
Published electronically: October 16, 2012
Keywords:Symmetric functions, Schur functions, tableaux, Schubert calculus
MSC (2010): Primary 05E05; Secondary 14N15

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. The poset of $k$-shapes
  • Chapter 3. Equivalence of paths in the poset of $k$-shapes
  • Chapter 4. Strips and tableaux for $k$-shapes
  • Chapter 5. Pushout of strips and row moves
  • Chapter 6. Pushout of strips and column moves
  • Chapter 7. Pushout sequences
  • Chapter 8. Pushouts of equivalent paths are equivalent
  • Chapter 9. Pullbacks
  • Appendix A. Tables of branching polynomials

Abstract


We 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. We define a symmetric function for each $k$-shape, and show that they expand positively in terms of dual $k$-Schur functions. We obtain an explicit combinatorial description of the expansion of an ungraded $k$-Schur function into $k+1$-Schur functions. As a corollary, we give a formula for the Schur expansion of an ungraded $k$-Schur function.

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