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arXiv math/01120222001-12-03math.QAmath.AGmath.CO

Grassmannian的量子上同调与全正性

Quantum Cohomology of Grassmannians and Total Positivity

Konstanze Rietsch

备注：: Revised version of ESI preprint 923. To appear in Duke Math J

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摘要：

给出了D. Peterson把Grassmannian的量子上同调环和$GL_n$的某个子簇的约化坐标环等同起来。然后构造了这个子簇的全正部分，并给出了Schubert基元在全正点上的值的闭公式。然后，我们使用开发的方法给一个新的证明公式的Vafa和Intriligator和Bertram的结构常数（Gromov-Witten不变量）。最后，我们利用这些Gromov-Witten不变量的正性证明了Schur多项式在单位根上的某些不等式。

英文摘要：

We give a proof of a result of D. Peterson's identifying the quantum cohomology ring of a Grassmannian with the reduced coordinate ring of a certain subvariety of $GL_n$. The totally positive part of this subvariety is then constructed and we give closed formulas for the values of the Schubert basis elements on the totally positive points. We then use the developed methods to give a new proof of a formula of Vafa and Intriligator and Bertram for the structure constants (Gromov--Witten invariants). Finally, we use the positivity of these Gromov--Witten invariants to prove certain inequalities for Schur polynomials at roots of unity.

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