Properties

Label 1.6.I.2.1a
  
Name \(\mathrm{SU}(2)\times E_2\)
Weight $1$
Degree $6$
Real dimension $6$
Components $2$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{SU}(2)\times\mathrm{SU}(2)_2\)
Component group \(C_2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$6$
Components:$2$
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:$C_2$
Order:$2$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 &0 & 0 \\0 & 0 & 0 & i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & -i \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$\mathrm{SU}(2)\times E_1$
Minimal supergroups:$\mathrm{SU}(2)\times E_6$, $\mathrm{SU}(2)\times E_4$, $\mathrm{SU}(2)\times J(E_2)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $30$ $0$ $465$ $0$ $8806$ $0$ $187446$ $0$ $4319436$
$a_2$ $1$ $2$ $11$ $69$ $531$ $4628$ $44045$ $446288$ $4739171$ $52197030$ $592008193$ $6878489621$ $81554916013$
$a_3$ $1$ $0$ $14$ $0$ $1228$ $0$ $183420$ $0$ $35005460$ $0$ $7734397416$ $0$ $1884962825328$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $11$ $5$ $16$ $30$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $14$ $69$ $39$ $121$ $70$ $232$ $465$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $95$ $531$ $306$ $182$ $1019$ $600$ $2052$ $1205$ $4218$ $8806$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $794$ $4628$ $462$ $2692$ $1576$ $9389$ $5488$ $3222$ $19516$ $11399$ $41113$ $23954$ $87444$ $187446$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1228$ $7217$ $44045$ $4218$ $25565$ $14892$ $92418$ $8685$ $53652$ $31203$ $196554$ $113898$ $66130$ $421711$
$$ $243880$ $910978$ $525714$ $1978920$ $4319436$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&4&0&1&0&1&0&0&4\\0&3&0&2&0&8&0&0&10&0&3&0&7&17&0\\1&0&8&0&4&0&13&15&0&6&0&18&0&0&36\\0&2&0&7&0&12&0&0&12&0&3&0&20&26&0\\1&0&4&0&9&0&13&16&0&13&0&24&0&0&44\\0&8&0&12&0&38&0&0&48&0&20&0&60&92&0\\1&0&13&0&13&0&38&35&0&21&0&59&0&0&116\\4&0&15&0&16&0&35&52&0&30&0&60&0&0&132\\0&10&0&12&0&48&0&0&71&0&28&0&79&130&0\\1&0&6&0&13&0&21&30&0&30&0&46&0&0&96\\0&3&0&3&0&20&0&0&28&0&19&0&39&56&0\\1&0&18&0&24&0&59&60&0&46&0&119&0&0&224\\0&7&0&20&0&60&0&0&79&0&39&0&135&172&0\\0&17&0&26&0&92&0&0&130&0&56&0&172&269&0\\4&0&36&0&44&0&116&132&0&96&0&224&0&0&470\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&8&7&9&38&38&52&71&30&19&119&135&269&470&209&231&463&334&97\end{bmatrix}$

Event probabilities

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