Properties

Label 1.6.G.2.1b
  
Name \(F_{a}\times \mathrm{SU}(2)\)
Weight $1$
Degree $6$
Real dimension $5$
Components $2$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)^2\times\mathrm{SU}(2)\)
Component group \(C_2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$5$
Components:$2$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)^2\times\mathrm{SU}(2)$
$\mathbb{R}$-dimension:$5$
Description:$\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}: A,B\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),C\in\mathrm{SU}(2)\right\}$ 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:$C_2$
Order:$2$
Abelian:yes
Generators:$\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$F\times \mathrm{SU}(2)$
Minimal supergroups:$F_{a,b}\times \mathrm{SU}(2)$${}^{\times 2}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $41$ $0$ $620$ $0$ $11739$ $0$ $255885$ $0$ $6115626$
$a_2$ $1$ $3$ $14$ $84$ $627$ $5493$ $53714$ $565512$ $6264757$ $71996631$ $850720122$ $10275024576$ $126341748135$
$a_3$ $1$ $0$ $18$ $0$ $1460$ $0$ $228960$ $0$ $47862780$ $0$ $11549742624$ $0$ $3041666585760$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $14$ $8$ $22$ $41$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $18$ $84$ $50$ $153$ $94$ $303$ $620$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $118$ $627$ $374$ $228$ $1251$ $754$ $2598$ $1560$ $5490$ $11739$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $944$ $5493$ $568$ $3246$ $1936$ $11553$ $6832$ $4062$ $24756$ $14590$ $53572$ $31458$ $116753$ $255885$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1460$ $8628$ $53714$ $5100$ $31346$ $18384$ $116417$ $10828$ $67812$ $39642$ $254651$ $147900$ $86184$ $560290$
$$ $324506$ $1238391$ $715386$ $2747535$ $6115626$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&1&0&2&3&0&1&0&2&0&0&4\\0&4&0&4&0&10&0&0&10&0&5&0&11&17&0\\2&0&9&0&7&0&15&17&0&8&0&23&0&0&40\\0&4&0&6&0&14&0&0&16&0&7&0&21&29&0\\1&0&7&0&11&0&18&17&0&12&0&34&0&0&52\\0&10&0&14&0&47&0&0&54&0&30&0&78&110&0\\2&0&15&0&18&0&42&42&0&29&0&76&0&0&140\\3&0&17&0&17&0&42&53&0&30&0&82&0&0&160\\0&10&0&16&0&54&0&0&76&0&35&0&107&155&0\\1&0&8&0&12&0&29&30&0&28&0&57&0&0&120\\0&5&0&7&0&30&0&0&35&0&26&0&58&78&0\\2&0&23&0&34&0&76&82&0&57&0&164&0&0&296\\0&11&0&21&0&78&0&0&107&0&58&0&170&234&0\\0&17&0&29&0&110&0&0&155&0&78&0&234&349&0\\4&0&40&0&52&0&140&160&0&120&0&296&0&0&620\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&9&6&11&47&42&53&76&28&26&164&170&349&620&273&278&592&456&115\end{bmatrix}$

Event probabilities

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