等变Schubert类通过矩映射退化的正性
Positivity of Equivariant Schubert Classes Through Moment Map Degeneration
摘要：
对于具有典型环面作用的旗流形$M=G/B$，$T-$等变上同调由等变Schubert类生成，Weyl群$W$的每个元素$u$都有一个类$τ_u$.这些类是由它们对不动点集$M^T \simeq W$的限制所决定的，并且这些限制是在单根中具有非负整数系数的多项式。本文的主要结果是一个计算A、B、C型$τ_u(v)$的正公式。为了得到这个公式，我们确定$G/B$与一个通用的共伴随轨道，并使用戈尔丁和托尔曼的结果来计算$τ_u(v)$的诱导矩映射。我们的公式，作为一个总和的贡献，某些最大的上升链从$u$ $v$，遵循系统退化的矩映射，对应于退化的共伴随轨道。在类型A中，我们证明了我们的公式与Billey在\cite{Bi}中宣布的公式是明显等价的，但在类型C中，这两个公式是不等价的.
英文摘要：
For a flag manifold $M=G/B$ with the canonical torus action, the $T-$equivariant cohomology is generated by equivariant Schubert classes, with one class $τ_u$ for every element $u$ of the Weyl group $W$. These classes are determined by their restrictions to the fixed point set $M^T \simeq W$, and the restrictions are polynomials with nonnegative integer coefficients in the simple roots. The main result of this article is a positive formula for computing $τ_u(v)$ in types A, B, and C. To obtain this formula we identify $G/B$ with a generic co-adjoint orbit and use a result of Goldin and Tolman to compute $τ_u(v)$ in terms of the induced moment map. Our formula, given as a sum of contributions of certain maximal ascending chains from $u$ to $v$, follows from a systematic degeneration of the moment map, corresponding to degenerating the co-adjoint orbit. In type A we prove that our formula is manifestly equivalent to the formula announced by Billey in \cite{Bi}, but in type C, the two formulas are not equivalent.
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