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Quantum cohomology of G/P and homology of affine Grassmannian | | Thomas Lam
; Mark Shimozono
; | | Date: |
10 May 2007 | | Subject: | Algebraic Geometry (math.AG); Combinatorics (math.CO) | | Abstract: | Let G be a simple and simply-connected complex algebraic group, P subset G a
parabolic subgroup. We prove an unpublished result of D. Peterson which states
that the quantum cohomology QH^*(G/P) of a flag variety is, up to localization,
a quotient of the homology H_*(Gr_G) of the affine Grassmannian Gr_G of G. As
a consequence, all three-point genus zero Gromov-Witten invariants of $G/P$ are
identified with homology Schubert structure constants of H_*(Gr_G),
establishing the equivalence of the quantum and homology affine Schubert
calculi.
For the case G = B, we use the Mihalcea’s equivariant quantum Chevalley
formula for QH^*(G/B), together with relationships between the quantum Bruhat
graph of Brenti, Fomin and Postnikov and the Bruhat order on the affine Weyl
group. As byproducts we obtain formulae for affine Schubert homology classes in
terms of quantum Schubert polynomials. We give some applications in quantum
cohomology.
Our main results extend to the torus-equivariant setting. | | Source: | arXiv, arxiv.0705.1386 | | Services: | Forum | Review | PDF | Favorites | |
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