论文标题
$ a $ a $ peterson schubert cyculus的正式公式
A positive formula for type $A$ Peterson Schubert calculus
论文作者
论文摘要
彼得森（Peterson）的品种是量子共同体学，代表理论和组合学研究中出现的特殊Nilpotent Hessenberg品种。在类型$ a $中，Peterson品种$ y $是完整国旗品种$ fl（n; \ m mathbb c）$的子各种，并且在$ t $ $ t $的子组$ s \ cong \ mathbb c^*$ t $的动作下是不变的，其中$ t $是标准的（nank n nastary（nancy）（非组合）在$ fl（n; math; math; math; \ mathbb c）上。使用Harada和Tymoczko介绍的Peterson Schubert基础，通过将一组特定的Schubert类从$ H_T^*（fl（n; \ Mathbb c））限制为$ H_S^*（y）$，我们描述了Equivariant同胞的产品结构$ h_s^*（y）$。特别是，我们通过为其结构常数提供明确的积极组合公式来表明该产品在适当的意义上显然是积极的。我们的方法需要概括Vandermonde身份的二项式系数的新组合身份。
Peterson varieties are special nilpotent Hessenberg varieties that have appeared in the study of quantum cohomology, representation theory, and combinatorics. In type $A$, the Peterson variety $Y$ is a subvariety of the complete flag variety $Fl(n; \mathbb C)$, and is invariant under the action of a subgroup $S\cong \mathbb C^*$ of $T$, where $T$ is the standard (noncompact) torus acting on $Fl(n; \mathbb C)$. Using the Peterson Schubert basis introduced by Harada and Tymoczko obtained by restricting a specific set of Schubert classes from $H_T^*(Fl(n; \mathbb C))$ to $H_S^*(Y)$, we describe the product structure of the equivariant cohomology $H_{S}^*(Y)$. In particular, we show that the product is manifestly positive in an appropriate sense by providing an explicit positive combinatorial formula for its structure constants. Our method requires a new combinatorial identity of binomial coefficients that generalizes Vandermonde's identity.
