Properties

Label 1.6.G.2.1a
  
Name \(F_{ab}\times \mathrm{SU}(2)\)
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:$\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1\\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $32$ $0$ $535$ $0$ $10864$ $0$ $246204$ $0$ $6002436$
$a_2$ $1$ $3$ $13$ $75$ $567$ $5133$ $51639$ $553689$ $6197235$ $71607561$ $848452095$ $10261640865$ $126261833949$
$a_3$ $1$ $0$ $14$ $0$ $1316$ $0$ $223520$ $0$ $47638780$ $0$ $11539829280$ $0$ $3041202959520$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $13$ $6$ $17$ $32$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $14$ $75$ $40$ $127$ $76$ $256$ $535$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $98$ $567$ $322$ $192$ $1113$ $660$ $2346$ $1390$ $5025$ $10864$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $840$ $5133$ $496$ $2970$ $1748$ $10807$ $6328$ $3722$ $23374$ $13660$ $50973$ $29708$ $111776$ $246204$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1316$ $8076$ $51639$ $4724$ $29854$ $17376$ $112309$ $10148$ $65048$ $37782$ $246916$ $142702$ $82684$ $545487$
$$ $314552$ $1209600$ $696024$ $2690772$ $6002436$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&4&0&0&0&1&0&0&4\\0&3&0&3&0&8&0&0&9&0&4&0&10&17&0\\2&0&8&0&4&0&12&18&0&6&0&20&0&0&40\\0&3&0&5&0&12&0&0&15&0&6&0&20&29&0\\0&0&4&0&11&0&17&12&0&14&0&34&0&0&52\\0&8&0&12&0&42&0&0&52&0&28&0&76&108&0\\1&0&12&0&17&0&40&38&0&30&0&75&0&0&140\\4&0&18&0&12&0&38&58&0&26&0&76&0&0&160\\0&9&0&15&0&52&0&0&75&0&34&0&106&155&0\\0&0&6&0&14&0&30&26&0&30&0&60&0&0&120\\0&4&0&6&0&28&0&0&34&0&25&0&57&76&0\\1&0&20&0&34&0&75&76&0&60&0&161&0&0&296\\0&10&0&20&0&76&0&0&106&0&57&0&169&232&0\\0&17&0&29&0&108&0&0&155&0&76&0&232&347&0\\4&0&40&0&52&0&140&160&0&120&0&296&0&0&620\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&8&5&11&42&40&58&75&30&25&161&169&347&620&271&291&591&456&113\end{bmatrix}$

Event probabilities

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