固定层上的A型Hessenberg簇的维数
Dimensions of type $A$ Hessenberg varieties over a fixed sheet
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中文总结 AI 辅助
本文将极小层上Hessenberg簇维数相同的结论推广到任意层，证明了固定Hessenberg函数时，A型旗簇中同一层算子定义的Hessenberg簇维数一致。
中文摘要 AI 辅助
Hessenberg簇$\nmathcal{H}\n\mathrm{ess}(\mathsf{X},\mathbf{h})$是旗簇的子簇，由Hessenberg函数$\n\mathbf{h}: [n] \to [n]$和矩阵$\n\mathsf{X} \in \mathfrak{gl}_n(\mathbb{C})$参数化。近期Goldin与第二作者的工作证明了Hessenberg簇可平坦退化到极小层上的幂零Hessenberg簇，这意味着极小层上的所有Hessenberg簇维数相同。本文的主要结果将该维数结论推广到任意层：对固定的Hessenberg函数$\n\mathbf{h}:[n] \to [n]$，由李代数$\n\mathfrak{gl}_n(\mathbb{C})$同一层中的线性算子$\n\mathsf{X}$在A型旗簇上定义的所有Hessenberg簇$\n\mathcal{H}\n\mathrm{ess}(\mathsf{X},\mathbf{h})$，均具有相同的维数。
英文摘要
Hessenberg varieties $\mathcal{H}\mathrm{ess}(\mathsf{X},\mathbf{h})$ are subvarieties of the flag variety parameterized by a Hessenberg function $\mathbf{h}: [n] \to [n]$ and a matrix $\mathsf{X} \in \mathfrak{gl}_n(\mathbb{C})$. In recent work, Goldin and the second author showed the existence of flat degenerations of Hessenberg varieties to nilpotent Hessenberg varieties over the minimal sheet. This implies that all Hessenberg varieties over the minimal sheet have the same dimension. Our main result generalizes this dimension result to arbitrary sheets. Specifically, we prove that for a fixed Hessenberg function $\mathbf{h}:[n] \to [n]$, all Hessenberg varieties $\mathcal{H}\mathrm{ess}(\mathsf{X},\mathbf{h})$ defined in the type $A$ flag variety by linear operators $\mathsf{X}$ from the same sheet of the Lie algebra $\mathfrak{gl}_n(\mathbb{C})$ have the same dimension.
