Properties

Label 1.6.I.3.1a
  
Name \(\mathrm{SU}(2)\times E_3\)
Weight $1$
Degree $6$
Real dimension $6$
Components $3$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{SU}(2)\times\mathrm{SU}(2)_2\)
Component group \(C_3\)

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Invariants

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

Identity component

Name:$\mathrm{SU}(2)\times\mathrm{SU}(2)_2$
$\mathbb{R}$-dimension:$6$
Description:$\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&\overline{B}\end{bmatrix}: A,B\in\mathrm{SU}(2)\right\}$ Symplectic form:$\begin{bmatrix}J_2&0&0\\0&0&I_2\\0&-I_2&0\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$
Hodge circle:$\mathrm{diag}(u,\bar u, u,\bar u,\bar u,u)$

Component group

Name:$C_3$
Order:$3$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$\mathrm{SU}(2)\times E_1$
Minimal supergroups:$\mathrm{SU}(2)\times E_6$, $\mathrm{SU}(2)\times J(E_3)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $26$ $0$ $355$ $0$ $6258$ $0$ $128646$ $0$ $2917332$
$a_2$ $1$ $2$ $9$ $53$ $385$ $3238$ $30163$ $301934$ $3184749$ $34948064$ $395586301$ $4591357059$ $54406139191$
$a_3$ $1$ $0$ $12$ $0$ $872$ $0$ $124190$ $0$ $23426424$ $0$ $5161147212$ $0$ $1256936939016$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $9$ $5$ $14$ $26$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $12$ $53$ $31$ $93$ $56$ $178$ $355$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $73$ $385$ $226$ $136$ $735$ $438$ $1470$ $875$ $3008$ $6258$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $570$ $3238$ $338$ $1898$ $1122$ $6533$ $3844$ $2272$ $13514$ $7939$ $28359$ $16618$ $60144$ $128646$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $872$ $5013$ $30163$ $2946$ $17575$ $10282$ $63056$ $6033$ $36728$ $21447$ $133698$ $77710$ $45274$ $286155$
$$ $165942$ $616950$ $356958$ $1338156$ $2917332$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&2&0&1&0&1&0&0&2\\0&3&0&2&0&6&0&0&6&0&3&0&5&11&0\\1&0&6&0&4&0&9&11&0&4&0&12&0&0&24\\0&2&0&5&0&8&0&0&10&0&3&0&12&18&0\\1&0&4&0&7&0&9&10&0&7&0&18&0&0&28\\0&6&0&8&0&28&0&0&32&0&14&0&40&62&0\\1&0&9&0&9&0&26&25&0&17&0&39&0&0&78\\2&0&11&0&10&0&25&34&0&18&0&42&0&0&88\\0&6&0&10&0&32&0&0&47&0&16&0&57&86&0\\1&0&4&0&7&0&17&18&0&18&0&30&0&0&64\\0&3&0&3&0&14&0&0&16&0&15&0&27&38&0\\1&0&12&0&18&0&39&42&0&30&0&81&0&0&148\\0&5&0&12&0&40&0&0&57&0&27&0&87&116&0\\0&11&0&18&0&62&0&0&86&0&38&0&116&181&0\\2&0&24&0&28&0&78&88&0&64&0&148&0&0&318\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&6&5&7&28&26&34&47&18&15&81&87&181&318&137&153&299&218&57\end{bmatrix}$

Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.