Properties

Label 1.6.I.12.4a
  
Name \(\mathrm{SU}(2)\times J(E_6)\)
Weight $1$
Degree $6$
Real dimension $6$
Components $12$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{SU}(2)\times\mathrm{SU}(2)_2\)
Component group \(D_6\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$6$
Components:$12$
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:$D_6$
Order:$12$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{12}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{11} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$\mathrm{SU}(2)\times J(E_3)$${}^{\times 2}$, $\mathrm{SU}(2)\times E_6$, $\mathrm{SU}(2)\times J(E_2)$
Minimal supergroups:

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $14$ $0$ $175$ $0$ $2884$ $0$ $55314$ $0$ $1170972$
$a_2$ $1$ $2$ $7$ $32$ $196$ $1482$ $12835$ $121151$ $1213196$ $12698252$ $137670397$ $1536637324$ $17579215978$
$a_3$ $1$ $0$ $7$ $0$ $410$ $0$ $50735$ $0$ $8524390$ $0$ $1710928758$ $0$ $387502522044$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $7$ $3$ $8$ $14$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $7$ $32$ $17$ $48$ $29$ $89$ $175$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $38$ $196$ $112$ $69$ $350$ $213$ $691$ $420$ $1400$ $2884$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $272$ $1482$ $164$ $873$ $527$ $2913$ $1750$ $1059$ $5975$ $3585$ $12434$ $7434$ $26124$ $55314$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $410$ $2228$ $12835$ $1338$ $7587$ $4530$ $26496$ $2715$ $15731$ $9373$ $55738$ $33012$ $19617$ $118281$
$$ $69860$ $252658$ $148806$ $542640$ $1170972$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&0&0&0&0&0&1\\0&2&0&1&0&3&0&0&3&0&1&0&2&5&0\\1&0&4&0&1&0&4&6&0&1&0&5&0&0&10\\0&1&0&3&0&4&0&0&4&0&1&0&6&7&0\\0&0&1&0&5&0&5&3&0&5&0&9&0&0&12\\0&3&0&4&0&13&0&0&14&0&7&0&18&25&0\\0&0&4&0&5&0&13&9&0&7&0&18&0&0&31\\2&0&6&0&3&0&9&18&0&6&0&16&0&0&34\\0&3&0&4&0&14&0&0&20&0&8&0&22&33&0\\0&0&1&0&5&0&7&6&0&10&0&13&0&0&24\\0&1&0&1&0&7&0&0&8&0&8&0&12&15&0\\0&0&5&0&9&0&18&16&0&13&0&36&0&0&58\\0&2&0&6&0&18&0&0&22&0&12&0&38&44&0\\0&5&0&7&0&25&0&0&33&0&15&0&44&69&0\\1&0&10&0&12&0&31&34&0&24&0&58&0&0&115\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&3&5&13&13&18&20&10&8&36&38&69&115&51&57&102&76&23\end{bmatrix}$

Event probabilities

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