弧图运算与线性算子的不变子空间的退化。部件II
Operations on Arc Diagrams and Degenerations for Invariant Subspaces of Linear Operators. Part II
摘要：
对于一个划分$β$，记为$N_β$ Jordan型$β$的幂零线性算子。给定划分$β$，$γ$，我们研究了所有短正合序列$$ \mathcal E: 0\to N_α\to N_β\to N_γ\to 0$$的表示空间${}_2{\mathbb V}_γ^β$，其中$α$是每个部分至多为2的任何划分。由于$α$上的条件，序列$\mathcal E$的同构类型由弧图$Δ$给出;用${\mathbb V}_Δ$表示同构于$\mathcal E$的所有序列的${}_2{\mathbb V}_γ^β$的子集。因此，空间${}_2{\mathbb V}_γ^β$携带由类型${\mathbb V}_Δ$的子集给出的分层。我们计算每一层的维数，并表明层${\mathbb V}_Δ$的边界恰好由那些${\mathbb V}_{Δ'}$组成，其中$Δ'$是由五种可能类型的弧移动的非空序列{ \bf（A）-（E）}从$Δ$获得的。在[3]和[4]中研究了所有三个分区都固定的情况。在那里，类型{ \bf（A）-（D）}的弧移动足以描述${\mathbb V}_{α,γ}^β$中${\mathbb V}_Δ$的边界。我们的第五步{ \bf（E）}，“爆炸”，需要将一个弧分成两个极点，以允许分区$α$中的变化。
英文摘要：
For a partition $β$, denote by $N_β$ the nilpotent linear operator of Jordan type $β$. Given partitions $β$, $γ$, we investigate the representation space ${}_2{\mathbb V}_γ^β$ of all short exact sequences $$ \mathcal E: 0\to N_α\to N_β\to N_γ\to 0$$ where $α$ is any partition with each part at most 2. Due to the condition on $α$, the isomorphism type of a sequence $\mathcal E$ is given by an arc diagram $Δ$; denote by ${\mathbb V}_Δ$ the subset of ${}_2{\mathbb V}_γ^β$ of all sequences isomorphic to $\mathcal E$. Thus, the space ${}_2{\mathbb V}_γ^β$ carries a stratification given by the subsets of type ${\mathbb V}_Δ$. We compute the dimension of each stratum and show that the boundary of a stratum ${\mathbb V}_Δ$ consists exactly of those ${\mathbb V}_{Δ'}$ where $Δ'$ is obtained from $Δ$ by a non-empty sequence of arc moves of five possible types {\bf (A) -- (E)}. The case where all three partitions are fixed has been studied in [3] and [4]. There, arc moves of types {\bf (A) -- (D)} suffice to describe the boundary of a ${\mathbb V}_Δ$ in ${\mathbb V}_{α,γ}^β$. Our fifth move {\bf (E)}, "explosion", is needed to break up an arc into two poles to allow for changes in the partition $α$.
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