Quantum Schubert polynomials
Quantum Schubert polynomials
复制标题
量子舒伯特多项式
DOI：
10.1090/s0894-0347-97-00237-3
复制
发表时间：
1997
影响因子：
3.9
通讯作者：
A. Postnikov
中科院分区：
文献类型：
--
作者：
S. Fomin;S. Gelfand;A. Postnikov
其中In是由x1，.中的对称多项式生成的理想。，xn，而没有常数项。另一个，几何，描述的上同调环的旗帜流形是基于分解成舒伯特细胞的Fln。这些是由对称群Sn的元素w索引的偶数维单元。相应的上同调类α，称为舒伯特类，形成H*（Fln 2）中的加法基。为了使两个描述相关，人们想要确定2[xl，.，Xn]/In对应于同构（1.1）下的Schubert类。这是第一次在[2]（也见[8]）的一般情况下，任意复杂的半单李群。后来，Lascoux和Schiitzenberger [22]提出了这个理论的组合版本（对于类型A），通过引入舒伯特类的显着多项式代表，称为舒伯特多项式，记为Gw。最近，受到来自弦理论[31，30]的思想的激励，数学家为任何Kahler代数流形X定义了（小）量子上同调环QH*（X，2），这是经典上同调环的某种变形（参见，例如，[28，19，14]和其中的参考文献）。QH*（X，2）的加法结构与普通上同调的加法结构基本相同.特别地，QH*（Fln，Z）作为阿贝尔群与张量积H*（Fln，2）（0 Z[ql，...，qn-1]，其中qi是形式变量（变形参数）。然而，量子上同调的乘法结构是
where In is the ideal generated by symmetric polynomials in x1,... ,xn without constant term. Another, geometric, description of the cohomology ring of the flag manifold is based on the decomposition of Fln into Schubert cells. These are even-dimensional cells indexed by the elements w of the symmetric group Sn. The corresponding cohomology classes oa, called Schubert classes, form an additive basis in H* (Fln 2) . To relate the two descriptions, one would like to determine which elements of 2[xl, ... , Xn]/In correspond to the Schubert classes under the isomorphism (1.1). This was first done in [2] (see also [8]) for a general case of an arbitrary complex semisimple Lie group. Later, Lascoux and Schiitzenberger [22] came up with a combinatorial version of this theory (for the type A) by introducing remarkable polynomial representatives of the Schubert classes oa called Schubert polynomials and denoted Gw. Recently, motivated by ideas that came from the string theory [31, 30], mathematicians defined, for any Kahler algebraic manifold X, the (small) quantum cohomology ring QH* (X, 2), which is a certain deformation of the classical cohomology ring (see, e.g., [28, 19, 14] and references therein). The additive structure of QH* (X , 2) is essentially the same as that of ordinary cohomology. In particular, QH* (Fln , Z) is canonically isomorphic, as an abelian group, to the tensor product H* (Fln , 2) (0 Z[ql,..., qn-1], where the qi are formal variables (deformation parameters). The multiplicative structure of the quantum cohomology is however