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

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 $\mathrm{\Delta }$; denote by ${\mathbb{V}}_{\mathrm{\Delta }}$ the subset of ${}2{\mathbb V}{\gamma}^\beta$ofallsequencesisomorphicto$\mathcal E$. Thus, the space ${}_2{\mathbb{V}}_{\gamma }^{\beta }$ carries a stratification given by the subsets of type ${\mathbb{V}}_{\mathrm{\Delta }}$. We compute the dimension of each stratum and show that the boundary of a stratum ${\mathbb{V}}_{\mathrm{\Delta }}$ consists exactly of those ${\mathbb{V}}_{{\mathrm{\Delta }}^{\mathrm{\prime }}}$ where ${\mathrm{\Delta }}^{\mathrm{\prime }}$ is obtained from $\mathrm{\Delta }$ 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 ${\mathbb{V}}_{\mathrm{\Delta }}$ 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 $\alpha$.