利用截断对称函数的理想推广Tanisaki理想
Generalizing Tanisaki's ideal via ideals of truncated symmetric functions
摘要：
在多项式环$\mathbb{Z}[x_1,...,x_n]$中定义了一族理想$I_h$，它们由Hessenberg函数$h$（等价于Dyck路或充分划分）参数化.理想$I_h$在代数上推广了一个称为谷崎理想的理想族，谷崎理想用于称为施普林格理论的置换表示的几何构造中。为了定义$I_h$，我们使用变量${x_1,...,x_n}$的真子集中的多项式，这些多项式在相应的置换子群下是对称的。我们称这些多项式{ \em截断对称函数}，并显示不同类型的截断对称多项式的组合身份。然后证明了$I_h$的几个关键性质，包括如果$h>h'$在Dyck路上是自然偏序的，那么$I_{h} \subset I_{h'}$，并明确地构造了$I_h$的Gröbner基.我们使用的第二个家庭的理想$J_h$的一些索赔更容易看到，并证明，$I_h = J_h$。理想$J_h$出现在丁，马丁-雷纳，和Gasharov-雷纳的工作上的一个家庭的舒伯特品种称为分区品种。使用早期的工作的第一作者，目前的手稿证明，理想$I_h = J_h$推广Tanisaki理想代数和几何，从施普林格品种的一个家庭的幂零Hessenberg品种。
英文摘要：
We define a family of ideals $I_h$ in the polynomial ring $\mathbb{Z}[x_1,...,x_n]$ that are parametrized by Hessenberg functions $h$ (equivalently Dyck paths or ample partitions). The ideals $I_h$ generalize algebraically a family of ideals called the Tanisaki ideal, which is used in a geometric construction of permutation representations called Springer theory. To define $I_h$, we use polynomials in a proper subset of the variables ${x_1,...,x_n}$ that are symmetric under the corresponding permutation subgroup. We call these polynomials {\em truncated symmetric functions} and show combinatorial identities relating different kinds of truncated symmetric polynomials. We then prove several key properties of $I_h$, including that if $h>h'$ in the natural partial order on Dyck paths then $I_{h} \subset I_{h'}$, and explicitly construct a Gröbner basis for $I_h$. We use a second family of ideals $J_h$ for which some of the claims are easier to see, and prove that $I_h = J_h$. The ideals $J_h$ arise in work of Ding, Develin-Martin-Reiner, and Gasharov-Reiner on a family of Schubert varieties called partition varieties. Using earlier work of the first author, the current manuscript proves that the ideals $I_h = J_h$ generalize the Tanisaki ideals both algebraically and geometrically, from Springer varieties to a family of nilpotent Hessenberg varieties.
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