Representatives for unipotent classes and nilpotent orbits

Figures & data

Table 1. Eminent nilpotent representatives in classical types.

Type	Char.	Representation	Representative
An	Any	$V(n+1)$	$e_{{\alpha }_1}+\cdots +e_{{\alpha }_n}$
Bn; $n\ge 2$	$p\ne 2$	$V(2n+1)$	$e_{{\alpha }_1}+\cdots +e_{{\alpha }_n}$
	p = 2	$D(n+1)$	$e_{{\alpha }_1}+\cdots +e_{{\alpha }_n}$
Cn; $n\ge 2$	Any	V(2n)	$e_{{\alpha }_1}+\cdots +e_{{\alpha }_n}$
	p = 2	$W_l(n);$ $0<l<\frac{n}{2}$	$({\sum }_{i=1}^{n-1}{e}_{{\alpha }_i})+e_{2{\alpha }_l+\cdots +2{\alpha }_{n-1}+{\alpha }_n}$
Dn; $n\ge 4$	p = 2	$W_l(n);$ $\frac{n+1}{2}<l<n$	${\sum }_{i=1}^{n-1}{e}_{{\alpha }_i}+e_{{\beta }_l},$ where ${\beta }_l={\alpha }_n+{\sum }_{i=2l-n-1}^{n-2}{\alpha }_i$

Table 2. Eminent nilpotent representatives in exceptional types.

Type	Char.	Label	Representative
G2	Any	G2	$e_{\begin{array}{c}10\end{array}}+e_{\begin{array}{c}01\end{array}}$
	p = 3	${({\tilde{A}}_1)}_3$	$e_{\begin{array}{c}21\end{array}}+e_{\begin{array}{c}32\end{array}}$
F4	Any	F4	$e_{\begin{array}{c}1000\end{array}}+e_{\begin{array}{c}0100\end{array}}+e_{\begin{array}{c}0010\end{array}}+e_{\begin{array}{c}0001\end{array}}$
	p = 2	$F_4(a_2)$	$e_{\begin{array}{c}0001\end{array}}+e_{\begin{array}{c}0011\end{array}}+e_{\begin{array}{c}1100\end{array}}+e_{\begin{array}{c}0120\end{array}}$
	p = 2	${(C_3)}_2$	$e_{\begin{array}{c}0001\end{array}}+e_{\begin{array}{c}1110\end{array}}+e_{\begin{array}{c}0120\end{array}}+e_{\begin{array}{c}1222\end{array}}$
	p = 2	${({\tilde{A}}_2{A}_1)}_2$	$e_{\begin{array}{c}0111\end{array}}+e_{\begin{array}{c}1121\end{array}}+e_{\begin{array}{c}0122\end{array}}+e_{\begin{array}{c}1220\end{array}}$
E6	Any	E6	$e_{\begin{array}{c}10000\\ 0\end{array}}+e_{\begin{array}{c}00000\\ 1\end{array}}+e_{\begin{array}{c}01000\\ 0\end{array}}+e_{\begin{array}{c}00100\\ 0\end{array}}+e_{\begin{array}{c}00010\\ 0\end{array}}+e_{\begin{array}{c}00001\\ 0\end{array}}$
	Any	$E_6(a_1)$	$e_{\begin{array}{c}10000\\ 0\end{array}}+e_{\begin{array}{c}01000\\ 0\end{array}}+e_{\begin{array}{c}00010\\ 0\end{array}}+e_{\begin{array}{c}00001\\ 0\end{array}}+e_{\begin{array}{c}00100\\ 1\end{array}}+e_{\begin{array}{c}01100\\ 0\end{array}}$
	p = 2	$E_6(a_3)$	$e_{\begin{array}{c}10000\\ 0\end{array}}+e_{\begin{array}{c}00100\\ 1\end{array}}+e_{\begin{array}{c}01100\\ 0\end{array}}+e_{\begin{array}{c}00011\\ 0\end{array}}+e_{\begin{array}{c}00110\\ 1\end{array}}+e_{\begin{array}{c}01110\\ 1\end{array}}$
E7	Any	E7	$e_{\begin{array}{c}100000\\ 0\end{array}}+e_{\begin{array}{c}000000\\ 1\end{array}}+e_{\begin{array}{c}010000\\ 0\end{array}}+e_{\begin{array}{c}001000\\ 0\end{array}}+e_{\begin{array}{c}000100\\ 0\end{array}}+e_{\begin{array}{c}000010\\ 0\end{array}}+e_{\begin{array}{c}000001\\ 0\end{array}}$
	Any	$E_7(a_1)$	$e_{\begin{array}{c}100000\\ 0\end{array}}+e_{\begin{array}{c}010000\\ 0\end{array}}+e_{\begin{array}{c}000100\\ 0\end{array}}+e_{\begin{array}{c}000010\\ 0\end{array}}+e_{\begin{array}{c}000001\\ 0\end{array}}+e_{\begin{array}{c}001000\\ 1\end{array}}+e_{\begin{array}{c}011000\\ 0\end{array}}$
	Any	$E_7(a_2)$	$e_{\begin{array}{c}100000\\ 0\end{array}}+e_{\begin{array}{c}000000\\ 1\end{array}}+e_{\begin{array}{c}010000\\ 0\end{array}}+e_{\begin{array}{c}001000\\ 1\end{array}}+e_{\begin{array}{c}001100\\ 0\end{array}}+e_{\begin{array}{c}000110\\ 0\end{array}}+e_{\begin{array}{c}000011\\ 0\end{array}}$
	p = 2	$E_7(a_3)$	$e_{\begin{array}{c}100000\\ 0\end{array}}+e_{\begin{array}{c}000001\\ 0\end{array}}+e_{\begin{array}{c}001000\\ 1\end{array}}+e_{\begin{array}{c}011000\\ 0\end{array}}+e_{\begin{array}{c}000110\\ 0\end{array}}+e_{\begin{array}{c}001100\\ 1\end{array}}+e_{\begin{array}{c}011100\\ 1\end{array}}$
	p = 2	${(A_6)}_2$	$e_{\begin{array}{c}000110\\ 0\end{array}}+e_{\begin{array}{c}000011\\ 0\end{array}}+e_{\begin{array}{c}111000\\ 0\end{array}}+e_{\begin{array}{c}011000\\ 1\end{array}}+e_{\begin{array}{c}001100\\ 1\end{array}}+e_{\begin{array}{c}011100\\ 0\end{array}}+e_{\begin{array}{c}122100\\ 1\end{array}}$
E8	Any	E8	$e_{\begin{array}{c}1000000\\ 0\end{array}}+e_{\begin{array}{c}0000000\\ 1\end{array}}+e_{\begin{array}{c}0100000\\ 0\end{array}}+e_{\begin{array}{c}0010000\\ 0\end{array}}+e_{\begin{array}{c}0001000\\ 0\end{array}}+e_{\begin{array}{c}0000100\\ 0\end{array}}+e_{\begin{array}{c}0000010\\ 0\end{array}}+e_{\begin{array}{c}0000001\\ 0\end{array}}$
	Any	$E_8(a_1)$	$e_{\begin{array}{c}1000000\\ 0\end{array}}+e_{\begin{array}{c}0000000\\ 1\end{array}}+e_{\begin{array}{c}0001000\\ 0\end{array}}+e_{\begin{array}{c}0000100\\ 0\end{array}}+e_{\begin{array}{c}0000010\\ 0\end{array}}+e_{\begin{array}{c}0000001\\ 0\end{array}}+e_{\begin{array}{c}0010000\\ 1\end{array}}+e_{\begin{array}{c}0110000\\ 0\end{array}}$
	Any	$E_8(a_2)$	$e_{\begin{array}{c}1000000\\ 0\end{array}}+e_{\begin{array}{c}0000000\\ 1\end{array}}+e_{\begin{array}{c}0100000\\ 0\end{array}}+e_{\begin{array}{c}0000001\\ 0\end{array}}+e_{\begin{array}{c}0010000\\ 1\end{array}}+e_{\begin{array}{c}0011000\\ 0\end{array}}+e_{\begin{array}{c}0001100\\ 0\end{array}}+e_{\begin{array}{c}0000110\\ 0\end{array}}$
	p = 2	$E_8(a_3)$	$e_{\begin{array}{c}0000010\\ 0\end{array}}+e_{\begin{array}{c}0000001\\ 0\end{array}}+e_{\begin{array}{c}1100000\\ 0\end{array}}+e_{\begin{array}{c}0010000\\ 1\end{array}}+e_{\begin{array}{c}0110000\\ 0\end{array}}+e_{\begin{array}{c}0011000\\ 0\end{array}}+e_{\begin{array}{c}0001100\\ 0\end{array}}+e_{\begin{array}{c}0111000\\ 0\end{array}}$
	p = 2	$E_8(a_4)$	$e_{\begin{array}{c}1100000\\ 0\end{array}}+e_{\begin{array}{c}0010000\\ 1\end{array}}+e_{\begin{array}{c}0110000\\ 0\end{array}}+e_{\begin{array}{c}0011000\\ 0\end{array}}+e_{\begin{array}{c}0001100\\ 0\end{array}}+e_{\begin{array}{c}0000110\\ 0\end{array}}+e_{\begin{array}{c}0000011\\ 0\end{array}}+e_{\begin{array}{c}0111000\\ 0\end{array}}$
	p = 2	$E_8(a_5)$	$e_{\begin{array}{c}1100000\\ 0\end{array}}+e_{\begin{array}{c}0000110\\ 0\end{array}}+e_{\begin{array}{c}0000011\\ 0\end{array}}+e_{\begin{array}{c}0110000\\ 1\end{array}}+e_{\begin{array}{c}0011000\\ 1\end{array}}+e_{\begin{array}{c}0111000\\ 0\end{array}}+e_{\begin{array}{c}0011100\\ 0\end{array}}+e_{\begin{array}{c}0011100\\ 1\end{array}}$
	p = 2	${(D_7(a_1))}_2$	$e_{\begin{array}{c}0001000\\ 0\end{array}}+e_{\begin{array}{c}0000001\\ 0\end{array}}+e_{\begin{array}{c}1100000\\ 0\end{array}}+e_{\begin{array}{c}0011000\\ 0\end{array}}+e_{\begin{array}{c}0000011\\ 0\end{array}}+e_{\begin{array}{c}0011100\\ 1\end{array}}+e_{\begin{array}{c}0111100\\ 0\end{array}}+e_{\begin{array}{c}0121110\\ 1\end{array}}$
	p = 2	${(D_7)}_2$	$e_{\begin{array}{c}1000000\\ 0\end{array}}+e_{\begin{array}{c}0110000\\ 1\end{array}}+e_{\begin{array}{c}0011000\\ 1\end{array}}+e_{\begin{array}{c}0111000\\ 0\end{array}}+e_{\begin{array}{c}0011100\\ 0\end{array}}+e_{\begin{array}{c}0001110\\ 0\end{array}}+e_{\begin{array}{c}0000111\\ 0\end{array}}+e_{\begin{array}{c}1111111\\ 1\end{array}}$
	p = 2	${(D_5{A}_2)}_2$	$e_{\begin{array}{c}0000011\\ 0\end{array}}+e_{\begin{array}{c}0000111\\ 0\end{array}}+e_{\begin{array}{c}1111000\\ 1\end{array}}+e_{\begin{array}{c}1111100\\ 0\end{array}}+e_{\begin{array}{c}0121000\\ 1\end{array}}+e_{\begin{array}{c}0111100\\ 1\end{array}}+e_{\begin{array}{c}0011110\\ 1\end{array}}+e_{\begin{array}{c}0111110\\ 0\end{array}}$
	p = 3	${(A_7)}_3$	$e_{\begin{array}{c}0001110\\ 0\end{array}}+e_{\begin{array}{c}0000111\\ 0\end{array}}+e_{\begin{array}{c}1110000\\ 1\end{array}}+e_{\begin{array}{c}1111000\\ 0\end{array}}+e_{\begin{array}{c}0011100\\ 1\end{array}}+e_{\begin{array}{c}0111100\\ 0\end{array}}+e_{\begin{array}{c}0121000\\ 1\end{array}}+e_{\begin{array}{c}0011111\\ 0\end{array}}$

Table 3. Eminent unipotent representatives in classical types.

Type	Char.	Representation	Representative
An	Any	$V(n+1)$	$x_{{\alpha }_1}(1)\cdots x_{{\alpha }_n}(1)$
Bn; $n\ge 2$	$p\ne 2$	$V(2n+1)$	$x_{{\alpha }_1}(1)\cdots x_{{\alpha }_n}(1)$
	p = 2	$V(2n)+R$	$x_{{\alpha }_1}(1)\cdots x_{{\alpha }_n}(1)$
Cn; $n\ge 2$	Any	V(2n)	$x_{{\alpha }_1}(1)\cdots x_{{\alpha }_n}(1)$
Dn; $n\ge 4$	p = 2	$V(2n-2l+2)+V(2l-2)$	$({\prod }_{i=1}^{n-1}{x}_{{\alpha }_i}(1))x_{{\beta }_l}(1),$ where ${\beta }_l={\alpha }_n+{\sum }_{i=2l-n-1}^{n-2}{\alpha }_i$
		$\frac{n+1}{2}<l<n$

Table 4. Eminent unipotent representatives in exceptional types.

Type	Char.	Label
G2	Any	G2
	p = 3	${({\tilde{A}}_1)}_3$
F4	Any	F4
	p = 2	$F_4(a_2)$
	p = 2	${({\tilde{A}}_2{A}_1)}_2$
E6	Any	E6
	Any	$E_6(a_1)$
	p = 2	$E_6(a_3)$
E7	Any	E7
	Any	$E_7(a_1)$
	Any	$E_7(a_2)$
	p = 2	$E_7(a_3)$
E8	Any	E8
	Any	$E_8(a_1)$
	Any	$E_8(a_2)$
	p = 2	$E_8(a_3)$
	p = 2	$E_8(a_4)$
	p = 2	$E_8(a_5)$
	p = 2	${(D_5{A}_2)}_2$
	p = 3	${(A_7)}_3$

Table 5. Recognition data for nilpotent orbits and unipotent classes.

Type	Char.	Data for nilpotent orbits	Type	Char.	Data for unipotent classes
G2	$p\ne 3$	JBS on $V_{\text{min}}$	G2	$p\ne 3$	JBS on $V_{\text{min}}$
	p = 3	JBS on $\text{Lie}(G)$		p = 3	JBS on $\text{Lie}(G)$
F4	p = 2	JBS on $V_{\text{min}}$ and $\text{Lie}(G);$ DS	F4	$p\le 3$	JBS on $V_{\text{min}}$ and $\text{Lie}(G)$
	p = 3	JBS on $V_{\text{min}}$ and $\text{Lie}(G)$		$p\ge 5$	JBS on $V_{\text{min}}$
	$p\ge 5$	JBS on $V_{\text{min}}$	E6	$p\ne 3$	JBS on $V_{\text{min}}$
E6	p = 2	JBS on $\text{Lie}(G);$ DS		p = 3	JBS on $\text{Lie}(G)$
	p = 3	JBS on $\text{Lie}(G);$ ALG	E7	p = 2	JBS on $\text{Lie}(G);$ LS
	$p\ge 5$	JBS on $V_{\text{min}}$		p = 3	JBS on $V_{\text{min}}$ and $\text{Lie}(G)$
E7	p = 2	JBS on $\text{Lie}(G);$ DS; ALG		$p\ge 5$	JBS on $V_{\text{min}}$
	p = 3	JBS on $V_{\text{min}}$ and $\text{Lie}(G);$ DS	E8	p = 2	JBS on $\text{Lie}(G);$ LS
	$p\ge 5$	JBS on $V_{\text{min}}$		$p\ge 3$	JBS on $\text{Lie}(G)$
E8	p = 2	JBS on $\text{Lie}(G);$ DS; ALG; ALG’
	p = 3	JBS on $\text{Lie}(G);$ NDS; NIL
	p = 5	JBS on $\text{Lie}(G);$ DS
	$p\ge 7$	JBS on $\text{Lie}(G)$

Table 6. Maximal generalized subsystem subgroups.

G	Char.	M
An
Bn; $n\ge 2$	$p\ne 2$	$B_m{D}_{n-m};$ $1\le m\le n$
	p = 2	$B_m{B}_{n-m};$ $1\le m\le \frac{n}{2}$
	Any	Dn
Cn; $n\ge 2$	Any	$C_m{C}_{n-m};$ $1\le m\le \frac{n}{2}$
	p = 2	Dn
Dn; $n\ge 4$	Any	$D_m{D}_{n-m};$ $2\le m\le \frac{n}{2}$
	Any	$B_{n-1}$
	$p\ne 2$	$B_m{B}_{n-m-1};$ $1\le m\le \frac{n}{2}$

Table 7. Representatives for nilpotent orbits and unipotent classes of ${\text{Sp}}_4(k).$

Representative x	Char.	$V\downarrow k[x]$
0	Any	$W{(1)}^2$
$e_{\begin{array}{c}10\end{array}}$	Any	W(2)
$e_{\begin{array}{c}01\end{array}}$	Any	$W(1)+V(2)$
$e_{\begin{array}{c}01\end{array}}+e_{\begin{array}{c}21\end{array}}$	p = 2	$V{(2)}^2$
$e_{\begin{array}{c}10\end{array}}+e_{\begin{array}{c}01\end{array}}$	Any	V(4)

Table 8. Representatives for nilpotent orbits and unipotent classes of ${\text{Sp}}_6(k).$

Representative x	Char.	$V\downarrow k[x]$
0	Any	$W{(1)}^3$
$e_{\begin{array}{c}100\end{array}}+e_{\begin{array}{c}010\end{array}}$	Any	W(3)
$e_{\begin{array}{c}100\end{array}}$	Any	$W(2)+W(1)$
$e_{\begin{array}{c}100\end{array}}+e_{\begin{array}{c}001\end{array}}$	Any	$W(2)+V(2)$
$e_{\begin{array}{c}010\end{array}}+e_{\begin{array}{c}001\end{array}}$	Any	$W(1)+V(4)$
$e_{\begin{array}{c}001\end{array}}+e_{\begin{array}{c}221\end{array}}$	p = 2	$W(1)+V{(2)}^2$
$e_{\begin{array}{c}001\end{array}}$	Any	$W{(1)}^2+V(2)$
$e_{\begin{array}{c}010\end{array}}+e_{\begin{array}{c}001\end{array}}+e_{\begin{array}{c}221\end{array}}$	Any	$V(4)+V(2)$
$e_{\begin{array}{c}100\end{array}}+e_{\begin{array}{c}010\end{array}}+e_{\begin{array}{c}001\end{array}}$	Any	V(6)
$e_{\begin{array}{c}100\end{array}}+e_{\begin{array}{c}010\end{array}}+e_{\begin{array}{c}221\end{array}}$ (nilp. only)	p = 2	$W_1(3)$

Table 9. Representatives for nilpotent orbits and unipotent classes of ${\text{Sp}}_8(k).$

Representative x	Char.	$V\downarrow k[x]$
0	Any	$W{(1)}^4$
$e_{\begin{array}{c}1000\end{array}}+e_{\begin{array}{c}0100\end{array}}+e_{\begin{array}{c}0010\end{array}}$	Any	W(4)
$e_{\begin{array}{c}1000\end{array}}+e_{\begin{array}{c}0100\end{array}}$	Any	$W(3)+W(1)$
$e_{\begin{array}{c}1000\end{array}}+e_{\begin{array}{c}0100\end{array}}+e_{\begin{array}{c}0001\end{array}}$	Any	$W(3)+V(2)$
$e_{\begin{array}{c}1000\end{array}}+e_{\begin{array}{c}0010\end{array}}$	Any	$W{(2)}^2$
$e_{\begin{array}{c}1000\end{array}}$	Any	$W(2)+W{(1)}^2$
$e_{\begin{array}{c}1000\end{array}}+e_{\begin{array}{c}0001\end{array}}$	Any	$W(2)+W(1)+V(2)$
$e_{\begin{array}{c}0100\end{array}}+e_{\begin{array}{c}0001\end{array}}+e_{\begin{array}{c}2221\end{array}}$	p = 2	$W(2)+V{(2)}^2$
$e_{\begin{array}{c}1000\end{array}}+e_{\begin{array}{c}0010\end{array}}+e_{\begin{array}{c}0001\end{array}}$	Any	$W(2)+V(4)$
$e_{\begin{array}{c}0001\end{array}}$	Any	$W{(1)}^3+V(2)$
$e_{\begin{array}{c}0001\end{array}}+e_{\begin{array}{c}2221\end{array}}$	p = 2	$W{(1)}^2+V{(2)}^2$
$e_{\begin{array}{c}0010\end{array}}+e_{\begin{array}{c}0001\end{array}}$	Any	$W{(1)}^2+V(4)$
$e_{\begin{array}{c}0010\end{array}}+e_{\begin{array}{c}0001\end{array}}+e_{\begin{array}{c}2221\end{array}}$	Any	$W(1)+V(4)+V(2)$
$e_{\begin{array}{c}0100\end{array}}+e_{\begin{array}{c}0010\end{array}}+e_{\begin{array}{c}0001\end{array}}$	Any	$W(1)+V(6)$
$e_{\begin{array}{c}0010\end{array}}+e_{\begin{array}{c}0001\end{array}}+e_{\begin{array}{c}0221\end{array}}+e_{\begin{array}{c}2221\end{array}}$	p = 2	$V(4)+V{(2)}^2$
$e_{\begin{array}{c}1000\end{array}}+e_{\begin{array}{c}0010\end{array}}+e_{\begin{array}{c}0001\end{array}}+e_{\begin{array}{c}0221\end{array}}$	p = 2	$V{(4)}^2$
$e_{\begin{array}{c}0100\end{array}}+e_{\begin{array}{c}0010\end{array}}+e_{\begin{array}{c}0001\end{array}}+e_{\begin{array}{c}2221\end{array}}$	Any	$V(6)+V(2)$
$e_{\begin{array}{c}1000\end{array}}+e_{\begin{array}{c}0100\end{array}}+e_{\begin{array}{c}0010\end{array}}+e_{\begin{array}{c}0001\end{array}}$	Any	V(8)
$e_{\begin{array}{c}1000\end{array}}+e_{\begin{array}{c}0100\end{array}}+e_{\begin{array}{c}0010\end{array}}+e_{\begin{array}{c}2221\end{array}}$ (nilp. only)	p = 2	$W_1(4)$
$e_{\begin{array}{c}0100\end{array}}+e_{\begin{array}{c}0010\end{array}}+e_{\begin{array}{c}0221\end{array}}$ (nilp. only)	p = 2	$W_1(3)+W(1)$

Table 10. Representatives for unipotent classes of ${\text{SO}}_7(k).$

Representative x	Char.	$V\downarrow k[u]$
1	Any	$W{(1)}^3+V(1)$
		$W{(1)}^3+R$
$x_{\begin{array}{c}100\end{array}}(1)x_{\begin{array}{c}010\end{array}}(1)$	Any	$W(3)+V(1)$
		$W(3)+R$
$x_{\begin{array}{c}100\end{array}}(1)$	Any	$W(2)+W(1)+V(1)$
		$W(2)+W(1)+R$
$x_{\begin{array}{c}100\end{array}}(1)x_{\begin{array}{c}001\end{array}}(1)$	Any	$W(2)+V(3)$
		$W(2)+V(2)+R$
$x_{\begin{array}{c}001\end{array}}(1)$	Any	$W{(1)}^2+V(3)$
		$W{(1)}^2+V(2)+R$
$x_{\begin{array}{c}010\end{array}}(1)x_{\begin{array}{c}001\end{array}}(1)$	Any	$W(1)+V(5)$
		$W(1)+V(4)+R$
$x_{\begin{array}{c}100\end{array}}(1)x_{\begin{array}{c}122\end{array}}(1)$	p = 2	$W(1)+V{(2)}^2+R$
$x_{\begin{array}{c}100\end{array}}(1)x_{\begin{array}{c}010\end{array}}(1)x_{\begin{array}{c}012\end{array}}(1)$	p = 2	$V(4)+V(2)+R$
$x_{\begin{array}{c}100\end{array}}(1)x_{\begin{array}{c}010\end{array}}(1)x_{\begin{array}{c}001\end{array}}(1)$	Any	V(7)
		$V(6)+R$

Table 11. Representatives for nilpotent orbits of ${\text{SO}}_7(k).$

Representative x	Char.	$V\downarrow k[e]$
0	Any	$W{(1)}^3+V(1)$
		$W{(1)}^3+D(1)$
$e_{\begin{array}{c}100\end{array}}+e_{\begin{array}{c}010\end{array}}$	Any	$W(3)+V(1)$
		$W(3)+D(1)$
$e_{\begin{array}{c}100\end{array}}$	Any	$W(2)+W(1)+V(1)$
		$W(2)+W(1)+D(1)$
$e_{\begin{array}{c}100\end{array}}+e_{\begin{array}{c}001\end{array}}$	Any	$W(2)+V(3)$
		$W(2)+D(2)$
$e_{\begin{array}{c}001\end{array}}$	Any	$W{(1)}^2+V(3)$
		$W{(1)}^2+D(2)$
$e_{\begin{array}{c}010\end{array}}+e_{\begin{array}{c}001\end{array}}$	Any	$W(1)+V(5)$
		$W(1)+D(3)$
$e_{\begin{array}{c}100\end{array}}+e_{\begin{array}{c}122\end{array}}$	p = 2	$W_2(2)+W(1)+D(1)$
$e_{\begin{array}{c}100\end{array}}+e_{\begin{array}{c}010\end{array}}+e_{\begin{array}{c}012\end{array}}$	p = 2	$W_3(3)+D(1)$
$e_{\begin{array}{c}100\end{array}}+e_{\begin{array}{c}010\end{array}}+e_{\begin{array}{c}001\end{array}}$	Any	V(7)
		D(4)

Table 12. Representatives for nilpotent orbits and unipotent classes of ${\text{SO}}_8(k).$

Representative x	$V\downarrow k[e]$	$V\downarrow k[u]$
0	$W{(1)}^4$	$W{(1)}^4$
$e_{\begin{array}{c}100\\ 0\end{array}}+e_{\begin{array}{c}010\\ 0\end{array}}+e_{\begin{array}{c}001\\ 0\end{array}}$	W(4)	W(4)
$e_{\begin{array}{c}100\\ 0\end{array}}+e_{\begin{array}{c}010\\ 0\end{array}}+e_{\begin{array}{c}000\\ 1\end{array}}$	W(4)	W(4)
$e_{\begin{array}{c}100\\ 0\end{array}}+e_{\begin{array}{c}010\\ 0\end{array}}$	$W(3)+W(1)$	$W(3)+W(1)$
$e_{\begin{array}{c}100\\ 0\end{array}}+e_{\begin{array}{c}001\\ 0\end{array}}$	$W{(2)}^2$	$W{(2)}^2$
$e_{\begin{array}{c}100\\ 0\end{array}}+e_{\begin{array}{c}000\\ 1\end{array}}$	$W{(2)}^2$	$W{(2)}^2$
$e_{\begin{array}{c}100\\ 0\end{array}}$	$W(2)+W{(1)}^2$	$W(2)+W{(1)}^2$
$e_{\begin{array}{c}100\\ 0\end{array}}+e_{\begin{array}{c}001\\ 0\end{array}}+e_{\begin{array}{c}000\\ 1\end{array}}$	$W(2)+V(3)+V(1)$ $W(2)+W_2(2)$	$W(2)+V(3)+V(1)$ $W(2)+V{(2)}^2$
$e_{\begin{array}{c}001\\ 0\end{array}}+e_{\begin{array}{c}000\\ 1\end{array}}$	$W{(1)}^2+V(3)+V(1)$ $W{(1)}^2+W_2(2)$	$W{(1)}^2+V(3)+V(1)$ $W{(1)}^2+V{(2)}^2$
$e_{\begin{array}{c}010\\ 0\end{array}}+e_{\begin{array}{c}001\\ 0\end{array}}+e_{\begin{array}{c}000\\ 1\end{array}}$	$W(1)+V(5)+V(1)$ $W(1)+W_3(3)$	$W(1)+V(5)+V(1)$ $W(1)+V(4)+V(2)$
$e_{\begin{array}{c}100\\ 0\end{array}}+e_{\begin{array}{c}010\\ 0\end{array}}+e_{\begin{array}{c}001\\ 0\end{array}}+e_{\begin{array}{c}110\\ 1\end{array}}$	$V(5)+V(3)$ $W_3(4)$	$V(5)+V(3)$ $V{(4)}^2$
$e_{\begin{array}{c}100\\ 0\end{array}}+e_{\begin{array}{c}010\\ 0\end{array}}+e_{\begin{array}{c}001\\ 0\end{array}}+e_{\begin{array}{c}000\\ 1\end{array}}$	$V(7)+V(1)$ $W_4(4)$	$V(7)+V(1)$ $V(6)+V(2)$

Table 13. Representatives for nilpotent orbits and unipotent classes of ${\text{SO}}_{10}(k).$

Representative x	$V\downarrow k[e]$	$V\downarrow k[u]$
0	$W{(1)}^5$	$W{(1)}^5$
$e_{\begin{array}{c}1000\\ 0\end{array}}+e_{\begin{array}{c}0100\\ 0\end{array}}+e_{\begin{array}{c}0010\\ 0\end{array}}+e_{\begin{array}{c}0001\\ 0\end{array}}$	W(5)	W(5)
$e_{\begin{array}{c}1000\\ 0\end{array}}+e_{\begin{array}{c}0100\\ 0\end{array}}+e_{\begin{array}{c}0010\\ 0\end{array}}$	$W(4)+W(1)$	$W(4)+W(1)$
$e_{\begin{array}{c}1000\\ 0\end{array}}+e_{\begin{array}{c}0100\\ 0\end{array}}+e_{\begin{array}{c}0001\\ 0\end{array}}$	$W(3)+W(2)$	$W(3)+W(2)$
$e_{\begin{array}{c}1000\\ 0\end{array}}+e_{\begin{array}{c}0100\\ 0\end{array}}$	$W(3)+W{(1)}^2$	$W(3)+W{(1)}^2$
$e_{\begin{array}{c}1000\\ 0\end{array}}+e_{\begin{array}{c}0010\\ 0\end{array}}$	$W{(2)}^2+W(1)$	$W{(2)}^2+W(1)$
$e_{\begin{array}{c}1000\\ 0\end{array}}$	$W(2)+W{(1)}^3$	$W(2)+W{(1)}^3$
$e_{\begin{array}{c}1000\\ 0\end{array}}+e_{\begin{array}{c}0100\\ 0\end{array}}+e_{\begin{array}{c}0001\\ 0\end{array}}+e_{\begin{array}{c}0000\\ 1\end{array}}$	$W(3)+V(3)+V(1)$	$W(3)+V(3)+V(1)$
	$W(3)+W_2(2)$	$W(3)+V{(2)}^2$
$e_{\begin{array}{c}1000\\ 0\end{array}}+e_{\begin{array}{c}0001\\ 0\end{array}}+e_{\begin{array}{c}0000\\ 1\end{array}}$	$W(2)+W(1)+V(3)+V(1)$	$W(2)+W(1)+V(3)+V(1)$
	$W(2)+W(1)+W_2(2)$	$W(2)+W(1)+V{(2)}^2$
$e_{\begin{array}{c}0001\\ 0\end{array}}+e_{\begin{array}{c}0000\\ 1\end{array}}$	$W{(1)}^3+V(3)+V(1)$	$W{(1)}^3+V(3)+V(1)$
	$W{(1)}^3+W_2(2)$	$W{(1)}^3+V{(2)}^2$
$e_{\begin{array}{c}1000\\ 0\end{array}}+e_{\begin{array}{c}0010\\ 0\end{array}}+e_{\begin{array}{c}0001\\ 0\end{array}}+e_{\begin{array}{c}0000\\ 1\end{array}}$	$W(2)+V(5)+V(1)$	$W(2)+V(5)+V(1)$
	$W(2)+W_3(3)$	$W(2)+V(4)+V(2)$
$e_{\begin{array}{c}0010\\ 0\end{array}}+e_{\begin{array}{c}0001\\ 0\end{array}}+e_{\begin{array}{c}0000\\ 1\end{array}}$	$W{(1)}^2+V(5)+V(1)$	$W{(1)}^2+V(5)+V(1)$
	$W{(1)}^2+W_3(3)$	$W{(1)}^2+V(4)+V(2)$
$e_{\begin{array}{c}0100\\ 0\end{array}}+e_{\begin{array}{c}0010\\ 0\end{array}}+e_{\begin{array}{c}0001\\ 0\end{array}}+e_{\begin{array}{c}0110\\ 1\end{array}}$	$W(1)+V(5)+V(3)$	$W(1)+V(5)+V(3)$
	$W(1)+W_3(4)$	$W(1)+V{(4)}^2$
$e_{\begin{array}{c}0100\\ 0\end{array}}+e_{\begin{array}{c}0010\\ 0\end{array}}+e_{\begin{array}{c}0001\\ 0\end{array}}+e_{\begin{array}{c}0000\\ 1\end{array}}$	$W(1)+V(7)+V(1)$	$W(1)+V(7)+V(1)$
	$W(1)+W_4(4)$	$W(1)+V(6)+V(2)$
$e_{\begin{array}{c}1000\\ 0\end{array}}+e_{\begin{array}{c}0100\\ 0\end{array}}+e_{\begin{array}{c}0010\\ 0\end{array}}+e_{\begin{array}{c}0001\\ 0\end{array}}+e_{\begin{array}{c}0110\\ 1\end{array}}$	$V(7)+V(3)$	$V(7)+V(3)$
	$W_4(5)$	$W(1)+V(6)+V(4)$
$e_{\begin{array}{c}1000\\ 0\end{array}}+e_{\begin{array}{c}0100\\ 0\end{array}}+e_{\begin{array}{c}0010\\ 0\end{array}}+e_{\begin{array}{c}0001\\ 0\end{array}}+e_{\begin{array}{c}0000\\ 1\end{array}}$	$W(1)+V(9)+V(1)$	$W(1)+V(9)+V(1)$
	$W_5(5)$	$V(8)+V(2)$

Table 14. Chevalley basis for $\mathfrak{s}{\mathfrak{p}}_{2n}.$

$e_{{\epsilon }_i-{\epsilon }_j}=E_{i,j}-E_{n+j,n+i}\mathrm{}\text{for}\mathrm{}\text{all}\mathrm{}i\ne j,$

$e_{{\epsilon }_i+{\epsilon }_j}=E_{j,n+i}+E_{i,n+j}\mathrm{}\text{for}\mathrm{}\text{all}\mathrm{}i\ne j,$

$e_{-({\epsilon }_i+{\epsilon }_j)}=E_{n+j,i}+E_{n+i,j}\mathrm{}\text{for}\mathrm{}\text{all}\mathrm{}i\ne j,$

$e_{2{\epsilon }_i}=E_{i,n+i}\mathrm{}\text{for}\mathrm{}\text{all}\mathrm{}i,$

$e_{-2{\epsilon }_i}=E_{n+i,i}\mathrm{}\text{for}\mathrm{}\text{all}\mathrm{}i.$

Table 15. Type C_n, expressions for positive roots in terms of base Δ.

Root	$\alpha ={\sum }_{k=1}^n{c}_k{\alpha }_i$
${\epsilon }_i-{\epsilon }_j,$ $1\le i<j\le n$	$\sum _{i\le k\le j-1}{\alpha }_k$
${\epsilon }_i+{\epsilon }_j,$ $1\le i<j\le n$	$\sum _{i\le k\le j-1}{\alpha }_k+\sum _{j\le k\le n-1}2{\alpha }_k+{\alpha }_n$
$2{\epsilon }_i,$ $1\le i\le n$	$\sum _{i\le k\le n-1}2{\alpha }_k+{\alpha }_n$

Table 16. Chevalley basis of $\mathfrak{s}\mathfrak{o}(2n).$

$e_{{\epsilon }_i-{\epsilon }_j}=E_{i,j}-E_{n+j,n+i}\mathrm{}\text{for}\mathrm{}\text{all}\mathrm{}i\ne j,$

$e_{{\epsilon }_i+{\epsilon }_j}=E_{j,n+i}-E_{i,n+j}\mathrm{}\text{for}\mathrm{}\text{all}\mathrm{}i<j,$

$e_{-({\epsilon }_i+{\epsilon }_j)}=E_{n+i,j}-E_{n+j,i}\mathrm{}\text{for}\mathrm{}\text{all}\mathrm{}i<j.$

Table 17. Type D_n, expressions for positive roots in terms of base Δ.

Root	$\alpha ={\sum }_{k=1}^n{c}_k{\alpha }_i$
${\epsilon }_i-{\epsilon }_j,$ $1\le i<j\le n$	$\sum _{i\le k\le j-1}{\alpha }_k$
${\epsilon }_i+{\epsilon }_j,$ $1\le i<j\le n-1$	$\sum _{i\le k\le j-1}{\alpha }_k+\sum _{j\le k\le n-2}2{\alpha }_k+{\alpha }_{n-1}+{\alpha }_n$
${\epsilon }_i+{\epsilon }_n,$ $1\le i\le n-1$	$\sum _{i\le k\le n-2}{\alpha }_k+{\alpha }_n$

Table 18. Maximal subsystem overgroups of non-eminent distinguished unipotent elements in exceptional types.

Type	Char.	Label	Maximal subsystem overgroups
G2	Any	$G_2(a_1)$	A2, $A_1{\tilde{A}}_1$ $(p\ne 2),{\tilde{A}}_2$ (p = 3)
F4	Any	$F_4(a_1)$	B4, C4 (p = 2)
	$p\ne 2$	$F_4(a_2)$	$A_1{C}_3$
	Any	$F_4(a_3)$	B4, $A_2{\tilde{A}}_2$ $(p\ne 3),A_1{C}_3$ $(p\ne 2),$ C4 (p = 2)
	p = 2	${(C_3(a_1))}_2$	B4, C4
E6	$p\ne 2$	$E_6(a_3)$	$A_1{A}_5$
E7	$p\ne 2$	$E_7(a_3)$	$A_1{D}_6$
	Any	$E_7(a_4)$	$A_1{D}_6,$ A7 (p = 2)
	Any	$E_7(a_5)$	$A_1{D}_6$ $(p\ne 2),A_2{A}_5$ $(p\ne 3)$
E8	$p\ne 2$	$E_8(a_3)$	$A_1{E}_7$
	$p\ne 2$	$E_8(a_4)$	D8
	$p\ne 2$	$E_8(a_5)$	D8
	Any	$E_8(a_6)$	A8 $(p\ne 3),$ D8 $(p\ne 2)$
	Any	$E_8(a_7)$	$A_4{A}_4$ $(p\ne 5),A_2{E}_6$ $(p\ne 3),A_1{E}_7$ $(p\ne 2),$ D8 $(p\ne 2)$
	Any	$E_8(b_4)$	$A_1{E}_7,$ D8 (p = 2)
	Any	$E_8(b_5)$	$A_2{E}_6$ $(p\ne 3),A_1{E}_7$ $(p\ne 2)$
	Any	$E_8(b_6)$	$A_2{E}_6,$ D8 $(p\ne 2),$ A8 (p = 3)
	p = 2	${(D_7(a_1))}_2$	$A_1{E}_7,$ D8

Table 19. Maximal subsystem overalgebras of non-eminent distinguished nilpotent elements in exceptional types.

Type	Char.	Label	Maximal subsystem overalgebras
G2	Any	$G_2(a_1)$	A2, $A_1{\tilde{A}}_1$ $(p\ne 2)$
F4	Any	$F_4(a_1)$	B4
	$p\ne 2$	$F_4(a_2)$	$A_1{C}_3$
	Any	$F_4(a_3)$	B4, $A_2{\tilde{A}}_2$ $(p\ne 3),A_1{C}_3$ $(p\ne 2)$
	p = 2	${(C_3(a_1))}_2$	B4
E6	$p\ne 2$	$E_6(a_3)$	$A_1{A}_5$
E7	$p\ne 2$	$E_7(a_3)$	$A_1{D}_6$
	Any	$E_7(a_4)$	$A_1{D}_6$
	Any	$E_7(a_5)$	$A_1{D}_6$ $(p\ne 2),A_2{A}_5$ $(p\ne 3)$
E8	$p\ne 2$	$E_8(a_3)$	$A_1{E}_7$
	$p\ne 2$	$E_8(a_4)$	D8
	$p\ne 2$	$E_8(a_5)$	D8
	Any	$E_8(a_6)$	A8 $(p\ne 3),$ D8 $(p\ne 2)$
	Any	$E_8(a_7)$	$A_4{A}_4$ $(p\ne 5),A_2{E}_6$ $(p\ne 3),A_1{E}_7$ $(p\ne 2),$ D8 $(p\ne 2)$
	Any	$E_8(b_4)$	$A_1{E}_7$
	Any	$E_8(b_5)$	$A_2{E}_6$ $(p\ne 3),A_1{E}_7$ $(p\ne 2)$
	Any	$E_8(b_6)$	$A_2{E}_6,$ D8 $(p\ne 2)$

Table A1. G = F₄, p = 2.

Label	DS data
$F_4(a_1)$	$10,5,2,0$
$F_4(a_2)$	10, 3, 0
B3	14, 9, 6
$F_4(a_3)$	$14,7,3,0$
${(C_3(a_1))}_2$	$16,8,4,0$
${({\tilde{A}}_2{A}_1)}_2$	$16,8,3,0$
$C_3(a_1)$	$20,16,15,7,4,0$
${\tilde{A}}_2{A}_1$	$20,16,15,7,1,0$

Table A2. G = E₆, p = 2.

Label	DS data
D5	$12,8,6,2,0$
$E_6(a_3)$	12, 6, 0
D4	20, 17, 16
$D_4(a_1)$	$20,15,8,0$
$A_3{A}_1$	$24,21,20,14,12,4,0$
$A_2^2{A}_1$	$24,21,20,12,1,0$

Table A3. G = E₆, p = 3.

Label	ALG data
E6	31 (sc)/24 (ad)
$E_6(a_1)$	60 (sc)/29 (ad)

Table A4. G = E₇, p = 3.

Label	DS data
E6	15, 12, 9
$E_6(a_1)$	$15,9,4,0$
$D_6(a_1)$	19, 15
A6	19, 17

Table A5. G = E₇ (simply connected), p = 2.

Label	ALG data	Label	DS data
E7	62	$D_6(a_2)$	$26,21,20,12,2,0$
$E_7(a_1)$	67	$A_5{A}_1$	$26,21,20,12,1,0$
D6	58	${(A_3{A}_1)}^{(1)}$	$43,39,37,29,21,20,4,0$
$E_7(a_4)$	63	$A_2^2{A}_1$	$43,39,37,27,8,0$
$D_6(a_1)$	70
${(A_6)}_2$	75
$D_5{A}_1$	73
$E_7(a_5)$	91
D5	70
$E_6(a_3)$	84
$D_4{A}_1$	63
$A_3{A}_2{A}_1$	69
${(A_3{A}_2)}_2$	74
$D_4(a_1)A_1$	83
D4	66
$D_4(a_1)$	82

Table A6. G = E₇ (adjoint), p = 2.

Label	ALG data	Label	DS data
E7	56	$D_6(a_2)$	$26,21,20,12,2,0$
$E_7(a_1)$	58	$A_5{A}_1$	$26,21,20,12,1,0$
D6	52	$A_3{A}_2$	$39,35,26,23,7,0$
$E_7(a_4)$	54	D4	39, 35, 28
$D_6(a_1)$	56	$D_4(a_1)$	$39,35,28,26,14,0$
${(A_6)}_2$	57	${(A_3{A}_1)}^{(1)}$	$43,40,37,29,21,20,4,0$
$D_5{A}_1$	60	$A_2^2{A}_1$	$43,40,37,27,8,0$
$E_7(a_5)$	71
D5	42
$E_6(a_3)$	50
$A_4{A}_2$	54
$D_5(a_1)$	39
$D_4{A}_1$	57
$A_3{A}_2{A}_1$	58
${(A_3{A}_2)}_2$	60
$D_4(a_1)A_1$	65
$A_2^2$	56
A3	55

Table A7. G = E₈, p = 5.

Label	DS data
E8	10, 7, 0
$E_8(a_1)$	10, 1, 0

Table A8. G = E₈, p = 3.

Label	NDS data	Label	NIL data
E8	15, 18, 248	$E_6(a_3)A_1$	36
$E_8(a_1)$	16, 18, 248	$D_5(a_1)A_2$	43
$E_6{A}_1$	32, 33, 39
$E_8(b_6)$	$33,34,39,60,165,248$
$E_6(a_1)A_1$	$34,40,53,132,248$
${(A_7)}_3$	$33,35,47,105,248$
A7	36, 36, 36
E6	36, 37, 59
$E_6(a_1)$	39, 56, 60
$E_7(a_4)$	40, 47, 63
$A_6{A}_1$	38, 38, 38
$D_6(a_1)$	43, 49
A6	41, 41

Table A9. G = E₈, p = 2.

Label	ALG, ALG′ data	Label	ALG, ALG′ data	Label	DS data
E8	79, 74	$D_5{A}_2$	76, 67	$D_5{A}_1$	$44,39,37,29,19,18,2,0$
$E_8(a_1)$	79, 73	$E_8(a_7)$	102, 99	$E_7(a_5)$	$44,39,37,29,17,1,0$
$E_8(a_2)$	89, 85	$D_6(a_2)$	73, 71	$E_6(a_3)A_1$	$44,39,37,27,8,0$
$E_8(a_3)$	108, 103	$A_5{A}_1$	73, 58	$D_5(a_1)A_2$	$48,43,37,21,1,0$
$E_8(a_4)$	139, 136	D5	64, 60	$A_4{A}_3$	$48,43,37,19,1,0$
E7	68, 64	$E_6(a_3)$	70, 64	$A_3{A}_1^2$	$80,76,75,71,70,54,39,38,6,0$
$E_8(b_4)$	70, 63	${(D_4{A}_2)}_2$	64, 58	$A_2^2{A}_1^2$	$80,76,75,71,70,50,14,0$
$E_7(a_1)$	72, 69	$A_4{A}_2{A}_1$	64, 57
$E_8(a_5)$	79, 72	$D_5(a_1)A_1$	71, 64
$E_8(b_5)$	91, 86	$D_4{A}_2$	83, 73
${(D_7)}_2$	102, 98	$A_3^2$	88, 76
$E_8(a_6)$	107, 101	$D_4(a_1)A_2$	99, 72
D7	69, 65	$D_4{A}_1$	91, 88
$D_7(a_1)$	77, 70	$A_3{A}_2{A}_1$	92, 86
A7	85, 72	${(A_3{A}_2)}_2$	101, 95
$D_7(a_2)$	101, 93	$D_4(a_1)A_1$	109, 104
$E_7(a_2)$	61, 57	D4	88, 85
$E_6{A}_1$	61, 53	$D_4(a_1)$	92, 90
${(D_7(a_1))}_2$	69, 63	$A_3{A}_1$	94, 92
$E_7(a_3)$	82, 78	$2A_2{A}_1$	94, 87
D6	71, 68
${(D_5{A}_2)}_2$	72, 65
$E_7(a_4)$	74, 67
$A_6{A}_1$	77, 66
$D_6(a_1)$	82, 78
${(A_6)}_2$	85, 76