摘要翻译：
设G是单连通复代数群，P\子集G是抛物子群。本文证明了D.Peterson的一个未发表的结果，即标志簇的量子上同调QH^*(G/P)是G的仿射Grassmanian\Gr_G的同调H_*(Gr_G)的商，从而证明了G/P$的所有三点亏格零Gromov-Witten不变量都与H_*(Gr_G)的同调Schubert结构常数等价，从而建立了量子与同调仿射Schubert演算的等价性。对于G=B的情形，我们利用Mihalcea的QH^*(G/B)的等变量子Chevalley公式，以及Brenti、Fomin和Postnikov的Bruhat图与仿射Weyl群上Bruhat序之间的关系。作为副产品，我们用量子Schubert多项式得到了仿射Schubert同调类的公式。我们给出了量子上同调的一些应用。我们的主要结果推广到环面等变的设置。
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英文标题：
《Quantum cohomology of G/P and homology of affine Grassmannian》
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作者：
Thomas Lam and Mark Shimozono
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics        数学
二级分类：Combinatorics        组合学
分类描述：Discrete mathematics, graph theory, enumeration, combinatorial optimization, Ramsey theory, combinatorial game theory
离散数学，图论，计数，组合优化，拉姆齐理论，组合对策论
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英文摘要：
  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 We prove an unpublished result of D. Peterson which states that the quantum cohomology QH^*(G/P) of a flag variety is, up to localization, a quotient of the homology H_*(Gr_G) of the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.
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PDF链接：
https://arxiv.org/pdf/0705.1386