Properties

Label 1.6.I.2.1b
  
Name \(\mathrm{SU}(2)\times J(E_1)\)
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:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$\mathrm{SU}(2)\times E_1$
Minimal supergroups:$\mathrm{SU}(2)\times J(E_3)$, $\mathrm{SU}(2)\times J(E_2)$${}^{\times 2}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $30$ $0$ $465$ $0$ $8806$ $0$ $187446$ $0$ $4319436$
$a_2$ $1$ $3$ $13$ $75$ $547$ $4673$ $44171$ $446645$ $4740187$ $52199937$ $592016543$ $6878513689$ $81554985589$
$a_3$ $1$ $0$ $14$ $0$ $1228$ $0$ $183420$ $0$ $35005460$ $0$ $7734397416$ $0$ $1884962825328$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $13$ $6$ $17$ $30$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $14$ $75$ $40$ $123$ $72$ $234$ $465$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $98$ $547$ $310$ $182$ $1025$ $602$ $2056$ $1210$ $4223$ $8806$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $800$ $4673$ $468$ $2702$ $1582$ $9405$ $5496$ $3222$ $19528$ $11404$ $41123$ $23968$ $87458$ $187446$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1228$ $7236$ $44171$ $4224$ $25594$ $14904$ $92463$ $8700$ $53672$ $31218$ $196586$ $113918$ $66130$ $421741$
$$ $243894$ $911006$ $525756$ $1978962$ $4319436$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&4&0&0&0&1&0&0&4\\0&3&0&3&0&8&0&0&9&0&2&0&8&17&0\\2&0&8&0&4&0&12&18&0&6&0&16&0&0&36\\0&3&0&5&0&12&0&0&15&0&4&0&16&27&0\\0&0&4&0&9&0&15&12&0&12&0&26&0&0&44\\0&8&0&12&0&38&0&0&48&0&20&0&60&92&0\\1&0&12&0&15&0&36&36&0&26&0&59&0&0&116\\4&0&18&0&12&0&36&52&0&24&0&60&0&0&132\\0&9&0&15&0&48&0&0&67&0&26&0&84&129&0\\0&0&6&0&12&0&26&24&0&24&0&48&0&0&96\\0&2&0&4&0&20&0&0&26&0&19&0&43&54&0\\1&0&16&0&26&0&59&60&0&48&0&119&0&0&224\\0&8&0&16&0&60&0&0&84&0&43&0&127&174&0\\0&17&0&27&0&92&0&0&129&0&54&0&174&269&0\\4&0&36&0&44&0&116&132&0&96&0&224&0&0&470\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&8&5&9&38&36&52&67&24&19&119&127&269&470&201&231&445&324&79\end{bmatrix}$

Event probabilities

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