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Affine insertion and Pieri rules for the affine Grassmannian
About this Title
Thomas Lam, Department of Mathematics, Harvard University, Cambridge MA 02138 USA, 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:
2010; Volume 208, Number 977
ISBNs: 978-0-8218-4658-2 (print); 978-1-4704-0591-5 (online)
DOI: https://doi.org/10.1090/S0065-9266-10-00576-4
Published electronically: April 28, 2010
Keywords: Tableaux,
Robinson-Schensted insertion,
Schubert calculus,
Pieri formula,
affine Grassmannian
MSC: Primary 05E05, 14N15
Table of Contents
Chapters
- Introduction
- 1. Schubert Bases of $\mathrm {Gr}$ and Symmetric Functions
- 2. Strong Tableaux
- 3. Weak Tableaux
- 4. Affine Insertion and Affine Pieri
- 5. The Local Rule $\phi _{u,v}$
- 6. Reverse Local Rule
- 7. Bijectivity
- 8. Grassmannian Elements, Cores, and Bounded Partitions
- 9. Strong and Weak Tableaux Using Cores
- 10. Affine Insertion in Terms of Cores
Abstract
We study combinatorial aspects of the Schubert calculus of the affine Grassmannian $\textrm {Gr}$ associated with $SL(n,\mathbb {C})$. Our main results are:
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These results are obtained by interpreting the Schubert bases of $\textrm {Gr}$ combinatorially as generating functions of objects we call strong and weak tableaux, which are respectively defined using the strong and weak orders on the affine symmetric group. We define a bijection called affine insertion, generalizing the Robinson-Schensted Knuth correspondence, which sends certain biwords to pairs of tableaux of the same shape, one strong and one weak. Affine insertion offers a duality between the weak and strong orders which does not seem to have been noticed previously.
Our cohomology Pieri rule conjecturally extends to the affine flag manifold, and we give a series of related combinatorial conjectures.