Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula
Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula
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量子双舒伯特多项式、量子舒伯特多项式和 Vafa-Intriligator 公式
DOI：
10.1016/s0012-365x(99)00263-0
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发表时间：
1996
期刊：
影响因子：
--
通讯作者：
T. Maeno
中科院分区：
文献类型：
--
作者：
A. Kirillov;T. Maeno
我们研究了旗形的等变量子上同调代数的代数性质。引入并研究了等变量子上同调类的LasCoux-Schützenberger型代表的量子双Schubert多项式S̃w(x，y)。我们的方法是基于量子柯西恒等式。我们还定义量子舒伯特多项式S̃w(X)为一组单项式关于标量积的格拉姆-施密特正交化，由Grothendieck剩余定义。利用量子柯西恒等式证明了S̃w(X)=S̃w(x，y)|y=0，并由此得到了量子Schubert多项式S̃w(X)=∂w0(Y)S̃w0(x，y)|y=0的一个简单公式.我们还证明了旗流形的Vafa-Intriigator公式的高阶亏格模拟，并研究了量子剩余母函数。我们在对称群上引入了Ehresmann-Bruhat图，并建立了等变量子Pieri规则。
We study algebraic aspects of equivariant quantum cohomology algebra of the flag manifold. We introduce and study the quantum double Schubert polynomials S ̃ w (x, y), which are the Lascoux–Schützenberger type representatives of the equivariant quantum cohomology classes. Our approach is based on the quantum Cauchy identity. We define also quantum Schubert polynomials S ̃ w (x) as the Gram–Schmidt orthogonalization of some set of monomials with respect to the scalar product, defined by the Grothendieck residue. Using quantum Cauchy identity, we prove that S ̃ w (x)= S ̃ w (x, y)| y= 0 and as a corollary obtain a simple formula for the quantum Schubert polynomials S ̃ w (x)=∂ ww 0 (y) S ̃ w 0 (x, y)| y= 0. We also prove the higher genus analog of Vafa–Intriligator's formula for the flag manifolds and study the quantum residues generating function. We introduce the Ehresmann–Bruhat graph on the symmetric group and formulate the equivariant quantum Pieri rule.