论文标题
一些与置换性品种有关的黑森伯格品种的几何形状和组合
The Geometry and Combinatorics of Some Hessenberg Varieties Related to the Permutohedral Variety
论文作者
论文摘要
我们构建了一个具体的同构，从置换界品种到与Hessenberg函数相关的常规半imple hessenberg品种$ h _+（i）= i+1 $，$ 1 $，$ 1 \ le i \ le i \ le n-1 $。在定义同构的过程中，我们引入了一系列品种，我们称之为前体内品种。我们首先确定这些品种的曲折结构，并使用感谢您的品种理论计算欧拉的特征和贝蒂数。然后，我们描述了这些品种的共同体。我们还找到了一种自然的方式，可以使用STEMBRIDGE定义的代码编码同一个组的一维组成部分。应用我们构建的同构，我们还能够描述与Hessenberg函数相关的常规半神经赫森伯格品种的几何结构，由$ h_k =（2,3，\ cdots，k+1，k+1，n，n，\ cdots，n）$，$ 1 \ le K \ le K \ le n-3-3 $。特别是，我们能够写下各种各样的共同体学环。最后，我们确定这些Hessenberg品种上的置换组$ {\ frak s} _n $的点表示。
We construct a concrete isomorphism from the permutohedral variety to the regular semisimple Hessenberg variety associated to the Hessenberg function $h_+(i)=i+1$, $1\le i\le n-1$. In the process of defining the isomorphism, we introduce a sequence of varieties which we call the prepermutohedral varieties. We first determine the toric structure of these varieties and compute the Euler characteristics and the Betti numbers using the theory of toric varieties. Then, we describe the cohomology of these varieties. We also find a natural way to encode the one-dimensional components of the cohomology using the codes defined by Stembridge. Applying the isomorphisms we constructed, we are also able to describe the geometric structure of regular semisimple Hessenberg varieties associated to the Hessenberg function represented by $h_k= (2,3, \cdots, k+1, n,\cdots,n)$, $1\le k\le n-3$. In particular, we are able to write down the cohomology ring of the variety. Finally, we determine the dot representation of the permutation group ${\frak S}_n$ on these Hessenberg varieties.
