Root subsystems of C^{1}

sp(10), type $C^{1}_5$
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Semisimple subalgebras.

Page generated by the calculator project.

g: C^{1}_5. There are 36 table entries (= 34 larger than the Cartan subalgebra + the Cartan subalgebra + the full subalgebra).

Type k_{ss}: C^{1}_5 (Full subalgebra) Type C(k_{ss})_{ss}: 0	Type k_{ss}: C^{1}_4+A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: C^{1}_3+B^{1}_2 Type C(k_{ss})_{ss}: 0	Type k_{ss}: C^{1}_3+2A^{1}_1 Type C(k_{ss})_{ss}: 0
Type k_{ss}: 2B^{1}_2+A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: B^{1}_2+3A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 5A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{2}_4 Type C(k_{ss})_{ss}: 0
Type k_{ss}: C^{1}_4 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: A^{2}_3+A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: C^{1}_3+A^{2}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: C^{1}_3+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1
Type k_{ss}: A^{2}_2+B^{1}_2 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{2}_2+2A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 2B^{1}_2 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: B^{1}_2+A^{2}_1+A^{1}_1 Type C(k_{ss})_{ss}: 0
Type k_{ss}: B^{1}_2+2A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: A^{2}_1+3A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 4A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: A^{2}_3 Type C(k_{ss})_{ss}: A^{1}_1
Type k_{ss}: C^{1}_3 Type C(k_{ss})_{ss}: B^{1}_2	Type k_{ss}: A^{2}_2+A^{2}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{2}_2+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: B^{1}_2+A^{2}_1 Type C(k_{ss})_{ss}: A^{1}_1
Type k_{ss}: B^{1}_2+A^{1}_1 Type C(k_{ss})_{ss}: B^{1}_2	Type k_{ss}: 2A^{2}_1+A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{2}_1+2A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: 3A^{1}_1 Type C(k_{ss})_{ss}: B^{1}_2
Type k_{ss}: A^{2}_2 Type C(k_{ss})_{ss}: B^{1}_2	Type k_{ss}: B^{1}_2 Type C(k_{ss})_{ss}: C^{1}_3	Type k_{ss}: 2A^{2}_1 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: A^{2}_1+A^{1}_1 Type C(k_{ss})_{ss}: B^{1}_2
Type k_{ss}: 2A^{1}_1 Type C(k_{ss})_{ss}: C^{1}_3	Type k_{ss}: A^{2}_1 Type C(k_{ss})_{ss}: C^{1}_3	Type k_{ss}: A^{1}_1 Type C(k_{ss})_{ss}: C^{1}_4	Type k_{ss}: 0 (Cartan subalgebra) Type C(k_{ss})_{ss}: C^{1}_5

There are 19 parabolic, 9 pseudo-parabolic but not parabolic and 8 non pseudo-parabolic root subsystems.

The roots needed to generate the root subsystems are listed below.
["parabolic","0", []],
["parabolic","A^{1}_1", [[2, 2, 2, 2, 1]]],
["parabolic","A^{2}_1", [[1, 2, 2, 2, 1]]],
["parabolic","A^{2}_1+A^{1}_1", [[1, 2, 2, 2, 1], [0, 0, 2, 2, 1]]],
["parabolic","2A^{2}_1", [[1, 2, 2, 2, 1], [0, 0, 1, 2, 1]]],
["parabolic","B^{1}_2", [[2, 2, 2, 2, 1], [-1, 0, 0, 0, 0]]],
["parabolic","A^{2}_2", [[1, 2, 2, 2, 1], [0, -1, 0, 0, 0]]],
["parabolic","2A^{2}_1+A^{1}_1", [[1, 2, 2, 2, 1], [0, 0, 1, 2, 1], [0, 0, 0, 0, 1]]],
["parabolic","B^{1}_2+A^{2}_1", [[2, 2, 2, 2, 1], [-1, 0, 0, 0, 0], [0, 0, 1, 2, 1]]],
["parabolic","A^{2}_2+A^{1}_1", [[1, 2, 2, 2, 1], [0, -1, 0, 0, 0], [0, 0, 0, 2, 1]]],
["parabolic","A^{2}_2+A^{2}_1", [[1, 2, 2, 2, 1], [0, -1, 0, 0, 0], [0, 0, 0, 1, 1]]],
["parabolic","C^{1}_3", [[1, 2, 2, 2, 1], [0, -1, 0, 0, 0], [0, 0, -2, -2, -1]]],
["parabolic","A^{2}_3", [[1, 2, 2, 2, 1], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0]]],
["parabolic","A^{2}_2+B^{1}_2", [[1, 2, 2, 2, 1], [0, -1, 0, 0, 0], [0, 0, 0, 2, 1], [0, 0, 0, -1, 0]]],
["parabolic","C^{1}_3+A^{2}_1", [[1, 2, 2, 2, 1], [0, -1, 0, 0, 0], [0, 0, -2, -2, -1], [0, 0, 0, 1, 1]]],
["parabolic","A^{2}_3+A^{1}_1", [[1, 2, 2, 2, 1], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0], [0, 0, 0, 0, 1]]],
["parabolic","C^{1}_4", [[1, 2, 2, 2, 1], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0], [0, 0, 0, -2, -1]]],
["parabolic","A^{2}_4", [[1, 2, 2, 2, 1], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0], [0, 0, 0, -1, 0]]],
["parabolic","C^{1}_5", [[1, 2, 2, 2, 1], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, -1]]],
["pseudoParabolicNonParabolic","2A^{1}_1", [[2, 2, 2, 2, 1], [0, 2, 2, 2, 1]]],
["pseudoParabolicNonParabolic","A^{2}_1+2A^{1}_1", [[1, 2, 2, 2, 1], [0, 0, 2, 2, 1], [0, 0, 0, 2, 1]]],
["pseudoParabolicNonParabolic","B^{1}_2+A^{1}_1", [[2, 2, 2, 2, 1], [-1, 0, 0, 0, 0], [0, 0, 2, 2, 1]]],
["pseudoParabolicNonParabolic","B^{1}_2+A^{2}_1+A^{1}_1", [[2, 2, 2, 2, 1], [-1, 0, 0, 0, 0], [0, 0, 1, 2, 1], [0, 0, 0, 0, 1]]],
["pseudoParabolicNonParabolic","2B^{1}_2", [[2, 2, 2, 2, 1], [-1, 0, 0, 0, 0], [0, 0, 2, 2, 1], [0, 0, -1, 0, 0]]],
["pseudoParabolicNonParabolic","A^{2}_2+2A^{1}_1", [[1, 2, 2, 2, 1], [0, -1, 0, 0, 0], [0, 0, 0, 2, 1], [0, 0, 0, 0, 1]]],
["pseudoParabolicNonParabolic","C^{1}_3+A^{1}_1", [[1, 2, 2, 2, 1], [0, -1, 0, 0, 0], [0, 0, -2, -2, -1], [0, 0, 0, 2, 1]]],
["pseudoParabolicNonParabolic","C^{1}_3+B^{1}_2", [[1, 2, 2, 2, 1], [0, -1, 0, 0, 0], [0, 0, -2, -2, -1], [0, 0, 0, 2, 1], [0, 0, 0, -1, 0]]],
["pseudoParabolicNonParabolic","C^{1}_4+A^{1}_1", [[1, 2, 2, 2, 1], [0, -1, 0, 0, 0], [0, 0, -1, 0, 0], [0, 0, 0, -2, -1], [0, 0, 0, 0, 1]]],
["nonPseudoParabolic","3A^{1}_1", [[2, 2, 2, 2, 1], [0, 2, 2, 2, 1], [0, 0, 2, 2, 1]]],
["nonPseudoParabolic","4A^{1}_1", [[2, 2, 2, 2, 1], [0, 2, 2, 2, 1], [0, 0, 2, 2, 1], [0, 0, 0, 2, 1]]],
["nonPseudoParabolic","A^{2}_1+3A^{1}_1", [[1, 2, 2, 2, 1], [0, 0, 2, 2, 1], [0, 0, 0, 2, 1], [0, 0, 0, 0, 1]]],
["nonPseudoParabolic","B^{1}_2+2A^{1}_1", [[2, 2, 2, 2, 1], [-1, 0, 0, 0, 0], [0, 0, 2, 2, 1], [0, 0, 0, 2, 1]]],
["nonPseudoParabolic","5A^{1}_1", [[2, 2, 2, 2, 1], [0, 2, 2, 2, 1], [0, 0, 2, 2, 1], [0, 0, 0, 2, 1], [0, 0, 0, 0, 1]]],
["nonPseudoParabolic","B^{1}_2+3A^{1}_1", [[2, 2, 2, 2, 1], [-1, 0, 0, 0, 0], [0, 0, 2, 2, 1], [0, 0, 0, 2, 1], [0, 0, 0, 0, 1]]],
["nonPseudoParabolic","2B^{1}_2+A^{1}_1", [[2, 2, 2, 2, 1], [-1, 0, 0, 0, 0], [0, 0, 2, 2, 1], [0, 0, -1, 0, 0], [0, 0, 0, 0, 1]]],
["nonPseudoParabolic","C^{1}_3+2A^{1}_1", [[1, 2, 2, 2, 1], [0, -1, 0, 0, 0], [0, 0, -2, -2, -1], [0, 0, 0, 2, 1], [0, 0, 0, 0, 1]]]

LaTeX table with root subalgebra details.
\documentclass{article}
\usepackage{longtable, amssymb, lscape}
\begin{document}
Lie algebra type: $C^{1}_5$. There are 36 table entries (= 34 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
$C^{1}_5$&$0$&$25$&$0$&$0$&$5$&$0$&\\
$C^{1}_4+A^{1}_1$&$0$&$17$&$0$&$1$&$5$&$0$&\\
$C^{1}_3+B^{1}_2$&$0$&$13$&$0$&$0$&$5$&$0$&\\
$C^{1}_3+2A^{1}_1$&$0$&$11$&$0$&$2$&$5$&$0$&\\
$2B^{1}_2+A^{1}_1$&$0$&$9$&$0$&$1$&$5$&$0$&\\
$B^{1}_2+3A^{1}_1$&$0$&$7$&$0$&$3$&$5$&$0$&\\
$5A^{1}_1$&$0$&$5$&$0$&$5$&$5$&$0$&\\
$A^{2}_4$&$0$&$10$&$0$&$0$&$4$&$0$&\\
$C^{1}_4$&$A^{1}_1$&$16$&$1$&$0$&$4$&$1$&\\
$A^{2}_3+A^{1}_1$&$0$&$7$&$0$&$1$&$4$&$0$&\\
$C^{1}_3+A^{2}_1$&$0$&$10$&$0$&$1$&$4$&$0$&\\
$C^{1}_3+A^{1}_1$&$A^{1}_1$&$10$&$1$&$1$&$4$&$1$&\\
$A^{2}_2+B^{1}_2$&$0$&$7$&$0$&$0$&$4$&$0$&\\
$A^{2}_2+2A^{1}_1$&$0$&$5$&$0$&$2$&$4$&$0$&\\
$2B^{1}_2$&$A^{1}_1$&$8$&$1$&$0$&$4$&$1$&\\
$B^{1}_2+A^{2}_1+A^{1}_1$&$0$&$6$&$0$&$2$&$4$&$0$&\\
$B^{1}_2+2A^{1}_1$&$A^{1}_1$&$6$&$1$&$2$&$4$&$1$&\\
$A^{2}_1+3A^{1}_1$&$0$&$4$&$0$&$4$&$4$&$0$&\\
$4A^{1}_1$&$A^{1}_1$&$4$&$1$&$4$&$4$&$1$&\\
$A^{2}_3$&$A^{1}_1$&$6$&$1$&$0$&$3$&$1$&\\
$C^{1}_3$&$B^{1}_2$&$9$&$4$&$0$&$3$&$2$&\\
$A^{2}_2+A^{2}_1$&$0$&$4$&$0$&$1$&$3$&$0$&\\
$A^{2}_2+A^{1}_1$&$A^{1}_1$&$4$&$1$&$1$&$3$&$1$&\\
$B^{1}_2+A^{2}_1$&$A^{1}_1$&$5$&$1$&$1$&$3$&$1$&\\
$B^{1}_2+A^{1}_1$&$B^{1}_2$&$5$&$4$&$1$&$3$&$2$&\\
$2A^{2}_1+A^{1}_1$&$0$&$3$&$0$&$3$&$3$&$0$&\\
$A^{2}_1+2A^{1}_1$&$A^{1}_1$&$3$&$1$&$3$&$3$&$1$&\\
$3A^{1}_1$&$B^{1}_2$&$3$&$4$&$3$&$3$&$2$&\\
$A^{2}_2$&$B^{1}_2$&$3$&$4$&$0$&$2$&$2$&\\
$B^{1}_2$&$C^{1}_3$&$4$&$9$&$0$&$2$&$3$&\\
$2A^{2}_1$&$A^{1}_1$&$2$&$1$&$2$&$2$&$1$&\\
$A^{2}_1+A^{1}_1$&$B^{1}_2$&$2$&$4$&$2$&$2$&$2$&\\
$2A^{1}_1$&$C^{1}_3$&$2$&$9$&$2$&$2$&$3$&\\
$A^{2}_1$&$C^{1}_3$&$1$&$9$&$1$&$1$&$3$&\\
$A^{1}_1$&$C^{1}_4$&$1$&$16$&$1$&$1$&$4$&\\
$0$&$C^{1}_5$&$0$&$25$&$0$&$0$&$5$&\\
\end{longtable}
\end{document}