Operations on Arc Diagrams and Degenerations for Invariant Subspaces of Linear Operators. Part II

Abstract

For a partition $\beta$, denote by $N_\beta$ the nilpotent linear operator of Jordan type $\beta$. Given partitions $\beta$, $\gamma$, we investigate the representation space ${}_2{\mathbb V}_{\gamma}^\beta$ of all short exact sequences $$ \mathcal E: 0\to N_\alpha \to N_\beta \to N_\gamma \to 0$$ where $\alpha$ is any partition with each part at most 2. Due to the condition on $\alpha$, the isomorphism type of a sequence $\mathcal E$ is given by an arc diagram $\Delta$; denote by ${\mathbb V}_\Delta$ the subset of ${}_2{\mathbb V}_{\gamma}^\beta$ of all sequences isomorphic to $\mathcal E$. Thus, the space ${}_2{\mathbb V}_{\gamma}^\beta$ carries a stratification given by the subsets of type ${\mathbb V}_\Delta$. We compute the dimension of each stratum and show that the boundary of a stratum ${\mathbb V}_\Delta$ consists exactly of those ${\mathbb V}_{\Delta'}$ where $\Delta'$ is obtained from $\Delta$ 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 ${\mathbb V}_\Delta$ in ${\mathbb V}_{\alpha,\gamma}^\beta$. Our fifth move {\bf (E)}, "explosion", is needed to break up an arc into two poles to allow for changes in the partition $\alpha$.