AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
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: https://doi.org/10.1090/S0065-9266-2012-00655-1
Published electronically: October 16, 2012
Keywords: Symmetric functions,
Schur functions,
tableaux,
Schubert calculus
MSC: Primary 05E05; Secondary 14N15
Table of Contents
Chapters
- 1. Introduction
- 2. The poset of $k$-shapes
- 3. Equivalence of paths in the poset of $k$-shapes
- 4. Strips and tableaux for $k$-shapes
- 5. Pushout of strips and row moves
- 6. Pushout of strips and column moves
- 7. Pushout sequences
- 8. Pushouts of equivalent paths are equivalent
- 9. Pullbacks
- 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.- S. Assaf, and S. Billey, private communication.
- J. Bandlow, and J. Morse, Combinatorial expansions in $K$-theoretic bases, preprint arXiv:1106.1594.
- Jonah Blasiak, Cyclage, catabolism, and the affine Hecke algebra, Adv. Math. 228 (2011), no. 4, 2292–2351. MR 2836122, DOI 10.1016/j.aim.2011.07.006
- L.-C. Chen, Ph. D. Thesis, U. C. Berkeley, 2010.
- Mark Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), no. 4, 941–1006. MR 1839919, DOI 10.1090/S0894-0347-01-00373-3
- Thomas Lam, Affine Stanley symmetric functions, Amer. J. Math. 128 (2006), no. 6, 1553–1586. MR 2275911
- Thomas Lam, Schubert polynomials for the affine Grassmannian, J. Amer. Math. Soc. 21 (2008), no. 1, 259–281. MR 2350056, DOI 10.1090/S0894-0347-06-00553-4
- Thomas Lam, Affine Schubert classes, Schur positivity, and combinatorial Hopf algebras, Bull. Lond. Math. Soc. 43 (2011), no. 2, 328–334. MR 2781213, DOI 10.1112/blms/bdq110
- Thomas Lam, Luc Lapointe, Jennifer Morse, and Mark Shimozono, Affine insertion and Pieri rules for the affine Grassmannian, Mem. Amer. Math. Soc. 208 (2010), no. 977, xii+82. MR 2741963, DOI 10.1090/S0065-9266-10-00576-4
- Thomas Lam, Anne Schilling, and Mark Shimozono, $K$-theory Schubert calculus of the affine Grassmannian, Compos. Math. 146 (2010), no. 4, 811–852. MR 2660675, DOI 10.1112/S0010437X09004539
- L. Lapointe, A. Lascoux, and J. Morse, Tableau atoms and a new Macdonald positivity conjecture, Duke Math. J. 116 (2003), no. 1, 103–146. MR 1950481, DOI 10.1215/S0012-7094-03-11614-2
- L. Lapointe and J. Morse, Schur function analogs for a filtration of the symmetric function space, J. Combin. Theory Ser. A 101 (2003), no. 2, 191–224. MR 1961543, DOI 10.1016/S0097-3165(02)00012-2
- Luc Lapointe and Jennifer Morse, Tableaux on $k+1$-cores, reduced words for affine permutations, and $k$-Schur expansions, J. Combin. Theory Ser. A 112 (2005), no. 1, 44–81. MR 2167475, DOI 10.1016/j.jcta.2005.01.003
- Luc Lapointe and Jennifer Morse, A $k$-tableau characterization of $k$-Schur functions, Adv. Math. 213 (2007), no. 1, 183–204. MR 2331242, DOI 10.1016/j.aim.2006.12.005
- Luc Lapointe and Jennifer Morse, Quantum cohomology and the $k$-Schur basis, Trans. Amer. Math. Soc. 360 (2008), no. 4, 2021–2040. MR 2366973, DOI 10.1090/S0002-9947-07-04287-0
- L. Lapointe and M.E. Pinto, in preparation.
- I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144