Root subsystems of F^{1}

$F^{1}_4$
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Semisimple subalgebras.

Page generated by the calculator project.

g: F^{1}_4. There are 24 table entries (= 22 larger than the Cartan subalgebra + the Cartan subalgebra + the full subalgebra).

Type k_{ss}: F^{1}_4 (Full subalgebra) Type C(k_{ss})_{ss}: 0	Type k_{ss}: B^{1}_4 Type C(k_{ss})_{ss}: 0	Type k_{ss}: D^{1}_4 Type C(k_{ss})_{ss}: 0	Type k_{ss}: C^{1}_3+A^{1}_1 Type C(k_{ss})_{ss}: 0
Type k_{ss}: A^{1}_3+A^{2}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{2}_2+A^{1}_2 Type C(k_{ss})_{ss}: 0	Type k_{ss}: B^{1}_2+2A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 4A^{1}_1 Type C(k_{ss})_{ss}: 0
Type k_{ss}: C^{1}_3 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: B^{1}_3 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_3 Type C(k_{ss})_{ss}: A^{2}_1	Type k_{ss}: A^{2}_2+A^{1}_1 Type C(k_{ss})_{ss}: 0
Type k_{ss}: B^{1}_2+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: A^{1}_2+A^{2}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{2}_1+2A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 3A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1
Type k_{ss}: A^{2}_2 Type C(k_{ss})_{ss}: A^{1}_2	Type k_{ss}: B^{1}_2 Type C(k_{ss})_{ss}: 2A^{1}_1	Type k_{ss}: A^{1}_2 Type C(k_{ss})_{ss}: A^{2}_2	Type k_{ss}: A^{2}_1+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1
Type k_{ss}: 2A^{1}_1 Type C(k_{ss})_{ss}: B^{1}_2	Type k_{ss}: A^{2}_1 Type C(k_{ss})_{ss}: A^{1}_3	Type k_{ss}: A^{1}_1 Type C(k_{ss})_{ss}: C^{1}_3	Type k_{ss}: 0 (Cartan subalgebra) Type C(k_{ss})_{ss}: F^{1}_4

There are 12 parabolic, 8 pseudo-parabolic but not parabolic and 4 non pseudo-parabolic root subsystems. The roots needed to generate the root subsystems are listed below using the root indices in GAP.
["parabolic","0", []],
["parabolic","A^{1}_1", [24]],
["parabolic","A^{2}_1", [21]],
["parabolic","A^{2}_1+A^{1}_1", [21, 14]],
["parabolic","A^{1}_2", [24, 1]],
["parabolic","B^{1}_2", [24, 8]],
["parabolic","A^{2}_2", [21, 4]],
["parabolic","A^{1}_2+A^{2}_1", [24, 1, 7]],
["parabolic","A^{2}_2+A^{1}_1", [21, 4, 5]],
["parabolic","B^{1}_3", [24, 1, 6]],
["parabolic","C^{1}_3", [21, 4, 9]],
["parabolic","F^{1}_4", [24, 1, 6, 4]],
["pseudoParabolicNonParabolic","2A^{1}_1", [24, 16]],
["pseudoParabolicNonParabolic","A^{2}_1+2A^{1}_1", [21, 14, 1]],
["pseudoParabolicNonParabolic","B^{1}_2+A^{1}_1", [24, 8, 9]],
["pseudoParabolicNonParabolic","A^{1}_3", [24, 1, 2]],
["pseudoParabolicNonParabolic","A^{2}_2+A^{1}_2", [21, 4, 5, 2]],
["pseudoParabolicNonParabolic","A^{1}_3+A^{2}_1", [24, 1, 2, 4]],
["pseudoParabolicNonParabolic","C^{1}_3+A^{1}_1", [21, 4, 9, 2]],
["pseudoParabolicNonParabolic","B^{1}_4", [24, 1, 2, 3]],
["nonPseudoParabolic","3A^{1}_1", [24, 16, 9]],
["nonPseudoParabolic","4A^{1}_1", [24, 16, 9, 2]],
["nonPseudoParabolic","B^{1}_2+2A^{1}_1", [24, 8, 9, 2]],
["nonPseudoParabolic","D^{1}_4", [24, 1, 2, 9]]

The roots needed to generate the root subsystems are listed below.
["parabolic","0", []],
["parabolic","A^{1}_1", [[2, 3, 4, 2]]],
["parabolic","A^{2}_1", [[1, 2, 3, 2]]],
["parabolic","A^{2}_1+A^{1}_1", [[1, 2, 3, 2], [1, 2, 2, 0]]],
["parabolic","A^{1}_2", [[2, 3, 4, 2], [-1, 0, 0, 0]]],
["parabolic","B^{1}_2", [[2, 3, 4, 2], [-1, -1, -1, 0]]],
["parabolic","A^{2}_2", [[1, 2, 3, 2], [0, 0, 0, -1]]],
["parabolic","A^{1}_2+A^{2}_1", [[2, 3, 4, 2], [-1, 0, 0, 0], [0, 0, 1, 1]]],
["parabolic","A^{2}_2+A^{1}_1", [[1, 2, 3, 2], [0, 0, 0, -1], [1, 1, 0, 0]]],
["parabolic","B^{1}_3", [[2, 3, 4, 2], [-1, 0, 0, 0], [0, -1, -1, 0]]],
["parabolic","C^{1}_3", [[1, 2, 3, 2], [0, 0, 0, -1], [0, -1, -2, 0]]],
["parabolic","F^{1}_4", [[2, 3, 4, 2], [-1, 0, 0, 0], [0, -1, -1, 0], [0, 0, 0, -1]]],
["pseudoParabolicNonParabolic","2A^{1}_1", [[2, 3, 4, 2], [0, 1, 2, 2]]],
["pseudoParabolicNonParabolic","A^{2}_1+2A^{1}_1", [[1, 2, 3, 2], [1, 2, 2, 0], [1, 0, 0, 0]]],
["pseudoParabolicNonParabolic","B^{1}_2+A^{1}_1", [[2, 3, 4, 2], [-1, -1, -1, 0], [0, 1, 2, 0]]],
["pseudoParabolicNonParabolic","A^{1}_3", [[2, 3, 4, 2], [-1, 0, 0, 0], [0, -1, 0, 0]]],
["pseudoParabolicNonParabolic","A^{2}_2+A^{1}_2", [[1, 2, 3, 2], [0, 0, 0, -1], [1, 1, 0, 0], [0, -1, 0, 0]]],
["pseudoParabolicNonParabolic","A^{1}_3+A^{2}_1", [[2, 3, 4, 2], [-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, 0, 1]]],
["pseudoParabolicNonParabolic","C^{1}_3+A^{1}_1", [[1, 2, 3, 2], [0, 0, 0, -1], [0, -1, -2, 0], [0, 1, 0, 0]]],
["pseudoParabolicNonParabolic","B^{1}_4", [[2, 3, 4, 2], [-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0]]],
["nonPseudoParabolic","3A^{1}_1", [[2, 3, 4, 2], [0, 1, 2, 2], [0, 1, 2, 0]]],
["nonPseudoParabolic","4A^{1}_1", [[2, 3, 4, 2], [0, 1, 2, 2], [0, 1, 2, 0], [0, 1, 0, 0]]],
["nonPseudoParabolic","B^{1}_2+2A^{1}_1", [[2, 3, 4, 2], [-1, -1, -1, 0], [0, 1, 2, 0], [0, 1, 0, 0]]],
["nonPseudoParabolic","D^{1}_4", [[2, 3, 4, 2], [-1, 0, 0, 0], [0, -1, 0, 0], [0, -1, -2, 0]]]

LaTeX table with root subalgebra details.
\documentclass{article}
\usepackage{longtable, amssymb, lscape}
\begin{document}
Lie algebra type: $F^{1}_4$. There are 24 table entries (= 22 larger than the Cartan subalgebra + the Cartan subalgebra + the full subalgebra).
Let $\mathfrak g$ stand for the type of the regular subalgebra and $C(\mathfrak g)$ for the type of the centralizer. Let $r$ stand for the rank of $\mathfrak g$, let $r_c$ stand for the rank of the semisimple part of the centralizer, let $p$ stand for the number of positive roots of $\mathfrak g$, let $q$ stand for the number of positive roots of the centralizer, and let $m$ stand for the number of $A_1$ components (of all root lengths) of $\mathfrak g$. \begin{longtable}{cccccccc}
$\mathfrak g$ & $C(\mathfrak g)$& $p$ & $q$& $m$& $r$ & $c_r$ \\\endhead
$F^{1}_4$&$0$&$24$&$0$&$0$&$4$&$0$&\\
$B^{1}_4$&$0$&$16$&$0$&$0$&$4$&$0$&\\
$D^{1}_4$&$0$&$12$&$0$&$0$&$4$&$0$&\\
$C^{1}_3+A^{1}_1$&$0$&$10$&$0$&$1$&$4$&$0$&\\
$A^{1}_3+A^{2}_1$&$0$&$7$&$0$&$1$&$4$&$0$&\\
$A^{2}_2+A^{1}_2$&$0$&$6$&$0$&$0$&$4$&$0$&\\
$B^{1}_2+2A^{1}_1$&$0$&$6$&$0$&$2$&$4$&$0$&\\
$4A^{1}_1$&$0$&$4$&$0$&$4$&$4$&$0$&\\
$C^{1}_3$&$A^{1}_1$&$9$&$1$&$0$&$3$&$1$&\\
$B^{1}_3$&$0$&$9$&$0$&$0$&$3$&$0$&\\
$A^{1}_3$&$A^{2}_1$&$6$&$1$&$0$&$3$&$1$&\\
$A^{2}_2+A^{1}_1$&$0$&$4$&$0$&$1$&$3$&$0$&\\
$B^{1}_2+A^{1}_1$&$A^{1}_1$&$5$&$1$&$1$&$3$&$1$&\\
$A^{1}_2+A^{2}_1$&$0$&$4$&$0$&$1$&$3$&$0$&\\
$A^{2}_1+2A^{1}_1$&$0$&$3$&$0$&$3$&$3$&$0$&\\
$3A^{1}_1$&$A^{1}_1$&$3$&$1$&$3$&$3$&$1$&\\
$A^{2}_2$&$A^{1}_2$&$3$&$3$&$0$&$2$&$2$&\\
$B^{1}_2$&$2A^{1}_1$&$4$&$2$&$0$&$2$&$2$&\\
$A^{1}_2$&$A^{2}_2$&$3$&$3$&$0$&$2$&$2$&\\
$A^{2}_1+A^{1}_1$&$A^{1}_1$&$2$&$1$&$2$&$2$&$1$&\\
$2A^{1}_1$&$B^{1}_2$&$2$&$4$&$2$&$2$&$2$&\\
$A^{2}_1$&$A^{1}_3$&$1$&$6$&$1$&$1$&$3$&\\
$A^{1}_1$&$C^{1}_3$&$1$&$9$&$1$&$1$&$3$&\\
$0$&$F^{1}_4$&$0$&$24$&$0$&$0$&$4$&\\
\end{longtable}
\end{document}