Properties

Label 1.6.H.8.5a
  
Name \(H_{a,b,c}\)
Weight $1$
Degree $6$
Real dimension $3$
Components $8$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)^3\)
Component group \(C_2^3\)

Learn more

Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$3$
Components:$8$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)^3$
$\mathbb{R}$-dimension:$3$
Description:$\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}: A,B,C\in\mathrm{U}(1)\subseteq\mathrm{SU}(2)\right\}$ Symplectic form:$\begin{bmatrix}J_2&0&0\\0&J_2&0\\0&0&J_2\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$
Hodge circle:$u\mapsto\mathrm{diag}(u,\bar u, u, \bar u, u, \bar u)$

Component group

Name:$C_2^3$
Order:$8$
Abelian:yes
Generators:$\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0& 0 & 0 & 0 & 1 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 &0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$H_{a,b}$${}^{\times 3}$, $H_{a,bc}$${}^{\times 3}$, $H_{ab,bc}$
Minimal supergroups:

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $27$ $0$ $390$ $0$ $7455$ $0$ $170478$ $0$ $4382532$
$a_2$ $1$ $3$ $12$ $60$ $396$ $3348$ $33699$ $377541$ $4509780$ $56058108$ $715392207$ $9300665421$ $122612100939$
$a_3$ $1$ $0$ $13$ $0$ $867$ $0$ $148720$ $0$ $37140635$ $0$ $10629900288$ $0$ $3251219481048$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $12$ $6$ $15$ $27$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $13$ $60$ $33$ $96$ $60$ $189$ $390$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $75$ $396$ $228$ $141$ $756$ $459$ $1590$ $960$ $3420$ $7455$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $564$ $3348$ $342$ $1944$ $1161$ $7056$ $4158$ $2472$ $15444$ $9060$ $34170$ $19950$ $76125$ $170478$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $867$ $5184$ $33699$ $3051$ $19368$ $11262$ $74454$ $6594$ $42876$ $24828$ $166806$ $95730$ $55215$ $375720$
$$ $214935$ $849576$ $484596$ $1927044$ $4382532$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&3&0&0&0&0&0&0&2\\0&3&0&3&0&6&0&0&6&0&3&0&6&9&0\\2&0&7&0&3&0&8&12&0&3&0&12&0&0&22\\0&3&0&4&0&8&0&0&9&0&4&0&12&16&0\\0&0&3&0&9&0&12&6&0&9&0&24&0&0&30\\0&6&0&8&0&28&0&0&30&0&20&0&48&68&0\\1&0&8&0&12&0&25&21&0&18&0&48&0&0&86\\3&0&12&0&6&0&21&39&0&12&0&48&0&0&102\\0&6&0&9&0&30&0&0&45&0&21&0&66&99&0\\0&0&3&0&9&0&18&12&0&23&0&36&0&0&82\\0&3&0&4&0&20&0&0&21&0&19&0&39&52&0\\0&0&12&0&24&0&48&48&0&36&0&114&0&0&198\\0&6&0&12&0&48&0&0&66&0&39&0&114&159&0\\0&9&0&16&0&68&0&0&99&0&52&0&159&247&0\\2&0&22&0&30&0&86&102&0&82&0&198&0&0&450\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&7&4&9&28&25&39&45&23&19&114&114&247&450&222&237&502&432&119\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$1/2$$0$$0$$0$$0$$1/2$
$a_1=0$$1/8$$1/8$$0$$0$$0$$0$$1/8$
$a_3=0$$1/8$$1/8$$0$$0$$0$$0$$1/8$
$a_1=a_3=0$$1/8$$1/8$$0$$0$$0$$0$$1/8$