An example In this question, we further assume that $N = 4$ and $\ker(h - \operatorname{id}_V) = \{0\}$. For $j \in \{0, \ldots, 3\}$, we denote $V_j = \ker(h - \zeta^j \operatorname{id}_V)$. According to $11^\circ$b), the data of $u$ is equivalent to the data of the two linear maps $u_1 : V_1 \rightarrow V_2$ and $u_2 : V_2 \rightarrow V_3$ induced by $u$. a) Verify that $u^3 = 0$. b) Construct pairs $(u, h)$ that give rise to six different types of pairs of diagonal blocks $(J_r, D_{r,a})$ in the ``graded'' version of the decomposition theorem. c) Prove that the number of blocks of each type is determined by the data of the three dimensions $d_j = \dim V_j$ ($1 \leqslant j \leqslant 3$) and the three ranks $r_1 = \operatorname{rg} u_1$, $r_2 = \operatorname{rg} u_2$ and $r_{21} = \operatorname{rg}(u_2 \circ u_1)$.
\textbf{An example}\\
In this question, we further assume that $N = 4$ and $\ker(h - \operatorname{id}_V) = \{0\}$. For $j \in \{0, \ldots, 3\}$, we denote $V_j = \ker(h - \zeta^j \operatorname{id}_V)$. According to $11^\circ$b), the data of $u$ is equivalent to the data of the two linear maps $u_1 : V_1 \rightarrow V_2$ and $u_2 : V_2 \rightarrow V_3$ induced by $u$.
a) Verify that $u^3 = 0$.
b) Construct pairs $(u, h)$ that give rise to six different types of pairs of diagonal blocks $(J_r, D_{r,a})$ in the ``graded'' version of the decomposition theorem.
c) Prove that the number of blocks of each type is determined by the data of the three dimensions $d_j = \dim V_j$ ($1 \leqslant j \leqslant 3$) and the three ranks $r_1 = \operatorname{rg} u_1$, $r_2 = \operatorname{rg} u_2$ and $r_{21} = \operatorname{rg}(u_2 \circ u_1)$.