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Affine Insertion and Pieri Rules for the Affine Grassmannian
Thomas Lam, Harvard University, Cambridge, MA, Luc Lapointe, Universidad de Talca, Chile, Jennifer Morse, Drexel University, Philadelphia, PA, and Mark Shimozono, Virginia Polytechnic Institute and State University, Blacksburg, VA

Memoirs of the American Mathematical Society
2010; 82 pp; softcover
Volume: 208
ISBN-10: 0-8218-4658-2
ISBN-13: 978-0-8218-4658-2
List Price: US$71
Individual Members: US$42.60
Institutional Members: US$56.80
Order Code: MEMO/208/977
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The authors study combinatorial aspects of the Schubert calculus of the affine Grassmannian \({\rm Gr}\) associated with \(SL(n,\mathbb{C})\).Their main results are:

  • Pieri rules for the Schubert bases of \(H^*({\rm Gr})\) and \(H_*({\rm Gr})\), which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes.
  • A new combinatorial definition for \(k\)-Schur functions, which represent the Schubert basis of \(H_*({\rm Gr})\).
  • A combinatorial interpretation of the pairing \(H^*({\rm Gr})\times H_*({\rm Gr}) \rightarrow\mathbb Z\) induced by the cap product.

Table of Contents

  • Schubert bases of \(\mathrm{Gr}\) and symmetric functions
  • Strong tableaux
  • Weak tableaux
  • Affine insertion and affine Pieri
  • The local rule \(\phi_{u,v}\)
  • Reverse local rule
  • Bijectivity
  • Grassmannian elements, cores, and bounded partitions
  • Strong and weak tableaux using cores
  • Affine insertion in terms of cores
  • Bibliography
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