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Affine insertion and Pieri rules for the affine Grassmannian
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|Title: ||Affine insertion and Pieri rules for the affine Grassmannian|
|Authors: ||Morse, Jennifer|
|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
• 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) -->
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.|
|Appears in Collections:||Faculty Research and Publications (Mathematics)|
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