We consider the space $X=\bigwedge ^{3}\mathbb{C}^{6}$ of alternating senary 3-tensors, equipped with the natural action of the group $\operatorname{GL}_{6}$ of invertible linear transformations of $\mathbb{C}^{6}$. We describe explicitly the category of $\operatorname{GL}_{6}$-equivariant coherent ${\mathcal{D}}_{X}$-modules as the category of representations of a quiver with relations, which has finite representation type. We give a construction of the six simple equivariant ${\mathcal{D}}_{X}$-modules and give formulas for the characters of their underlying $\operatorname{GL}_{6}$-structures. We describe the (iterated) local cohomology groups with supports given by orbit closures, determining, in particular, the Lyubeznik numbers associated to the orbit closures.

中文翻译：

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交替 SENARY 3-张量的等变模块

我们考虑空间$X=\bigwedge ^{3}\mathbb{C}^{6}$交变 senary 3-张量，配备群的自然作用$\运营商名称{GL}_{6}$的可逆线性变换$\mathbb{C}^{6}$. 我们明确描述的类别$\运营商名称{GL}_{6}$-等变相干${\mathcal{D}}_{X}$-模块作为具有关系的箭袋的表示类别，它具有有限的表示类型。我们给出了六个简单等变式的构造${\mathcal{D}}_{X}$-模块并给出其底层特征的公式$\运营商名称{GL}_{6}$-结构。我们用轨道闭合给出的支持来描述（迭代的）局部上同调群，特别是确定与轨道闭合相关的 Lyubeznik 数。