Properties

Label 1.6.G.4.2a
  
Name \(F_{a,b}\times \mathrm{SU}(2)\)
Weight $1$
Degree $6$
Real dimension $5$
Components $4$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)^2\times\mathrm{SU}(2)\)
Component group \(C_2^2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$5$
Components:$4$
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:$C_2^2$
Order:$4$
Abelian:yes
Generators:$\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0& 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $26$ $0$ $355$ $0$ $6314$ $0$ $132804$ $0$ $3114474$
$a_2$ $1$ $3$ $12$ $60$ $384$ $3048$ $28259$ $289761$ $3169420$ $36202692$ $426523599$ $5144292717$ $63211096881$
$a_3$ $1$ $0$ $13$ $0$ $818$ $0$ $117360$ $0$ $24045182$ $0$ $5779849488$ $0$ $1521065376336$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $12$ $6$ $15$ $26$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $13$ $60$ $33$ $94$ $58$ $178$ $355$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $75$ $384$ $222$ $136$ $708$ $430$ $1434$ $870$ $2985$ $6314$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $542$ $3048$ $328$ $1788$ $1074$ $6190$ $3686$ $2211$ $13096$ $7775$ $28108$ $16618$ $60886$ $132804$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $818$ $4644$ $28259$ $2762$ $16500$ $9732$ $60384$ $5774$ $35342$ $20781$ $131274$ $76594$ $44870$ $287614$
$$ $167272$ $633654$ $367416$ $1402212$ $3114474$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&3&0&0&0&0&0&0&2\\0&3&0&3&0&6&0&0&6&0&2&0&5&9&0\\2&0&7&0&3&0&8&12&0&3&0&10&0&0&20\\0&3&0&4&0&8&0&0&9&0&3&0&10&15&0\\0&0&3&0&8&0&11&6&0&8&0&20&0&0&26\\0&6&0&8&0&26&0&0&28&0&16&0&40&56&0\\1&0&8&0&11&0&23&20&0&16&0&40&0&0&70\\3&0&12&0&6&0&20&34&0&11&0&38&0&0&80\\0&6&0&9&0&28&0&0&39&0&17&0&53&78&0\\0&0&3&0&8&0&16&11&0&17&0&30&0&0&60\\0&2&0&3&0&16&0&0&17&0&16&0&32&38&0\\0&0&10&0&20&0&40&38&0&30&0&88&0&0&148\\0&5&0&10&0&40&0&0&53&0&32&0&88&116&0\\0&9&0&15&0&56&0&0&78&0&38&0&116&178&0\\2&0&20&0&26&0&70&80&0&60&0&148&0&0&310\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&7&4&8&26&23&34&39&17&16&88&88&178&310&140&155&299&237&61\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$1/4$$0$$0$$0$$0$$1/4$
$a_1=0$$0$$0$$0$$0$$0$$0$$0$
$a_3=0$$0$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$0$$0$$0$$0$$0$$0$$0$