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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Schubert polynomials for the affine Grassmannian


Author: Thomas Lam
Journal: J. Amer. Math. Soc. 21 (2008), 259-281
MSC (2000): Primary 05E05; Secondary 14N15
Published electronically: October 18, 2006
MathSciNet review: 2350056
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Abstract: Confirming a conjecture of Mark Shimozono, we identify polynomial representatives for the Schubert classes of the affine Grassmannian as the $ k$-Schur functions in homology and affine Schur functions in cohomology. The results are obtained by connecting earlier combinatorial work of ours to certain subalgebras of Kostant and Kumar's nilHecke ring and to work of Peterson on the homology of based loops on a compact group. As combinatorial corollaries, we settle a number of positivity conjectures concerning $ k$-Schur functions, affine Stanley symmetric functions and cylindric Schur functions.


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Additional Information

Thomas Lam
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: tfylam@math.harvard.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-06-00553-4
PII: S 0894-0347(06)00553-4
Keywords: Schubert polynomials, symmetric functions, Schubert calculus, affine Grassmannian
Received by editor(s): April 7, 2006
Published electronically: October 18, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.