Hessenberg varieties are not pure dimensional
Hessenberg varieties are not pure dimensional
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Hessenberg 品种不是纯维度的
DOI：
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发表时间：
2006
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影响因子：
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通讯作者：
Julianna Tymoczko
中科院分区：
文献类型：
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作者：
Julianna Tymoczko
我们研究了由某些线性条件定义的旗形品种的一系列子品种，称为海森伯格品种。我们将它们与舒伯特的变种进行比较。我们证明了一些Schubert簇可以实现为Hessenberg簇，反之亦然。我们的证明明确确定这些舒伯特品种的排列，并计算其尺寸。
我们用这个来回答一个悬而未决的问题，证明海森伯格品种并不总是纯维。我们给出的例子，既不半单也不幂零海森堡品种需要纯，后者是连接的，非纯维海森堡品种。我们的方法要求我们推广Hessenberg簇的定义。
We study a family of subvarieties of the flag variety defined by certain linear conditions, called Hessenberg varieties. We compare them to Schubert varieties. We prove that some Schubert varieties can be realized as Hessenberg varieties and vice versa. Our proof explicitly identifies these Schubert varieties by their permutation and computes their dimension.
We use this to answer an open question by proving that Hessenberg varieties are not always pure dimensional. We give examples that neither semisimple nor nilpotent Hessenberg varieties need be pure; the latter are connected, non-pure-dimensional Hessenberg varieties. Our methods require us to generalize the definition of Hessenberg varieties.