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Please use this identifier to cite or link to this item: http://hdl.handle.net/1860/1945

Title: Affine insertion and Pieri rules for the affine Grassmannian
Authors: Morse, Jennifer
Lam, Thomas
Lapointe, Luc
Shimozono, Mark
Issue Date: 4-Sep-2006
Citation: Retrieved October 30, 2007 from http://xxx.lanl.gov/find/grp_math/1/au:+Morse_J/0/1/0/all/0/1.
Abstract: We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n, C). Our main results are: • Pieri rules for the Schubert bases of H∗(Gr) and H∗(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∗(Gr). • A combinatorial interpretation of the pairing H∗(Gr)×H∗(Gr) --> Z. These results are obtained by interpreting the Schubert bases of 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.
URI: http://lanl.arxiv.org/abs/math/0609110v2
http://hdl.handle.net/1860/1945
Appears in Collections:Faculty Research and Publications (Mathematics)

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