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Properties

Label 1.6.H.2.1b
  
Name \(H_{ab}\)
Weight $1$
Degree $6$
Real dimension $3$
Components $2$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)^3\)
Component group \(C_2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$3$
Components:$2$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)^3$
$\mathbb{R}$-dimension:$3$
Description:$\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}: A,B,C\in\mathrm{U}(1)\subseteq\mathrm{SU}(2)\right\}$ 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:$H$
Minimal supergroups:$H_{at}$, $H_{act}$, $H_{a,b}$, $H_{a,bc}$, $H_{ab,bc}$${}^{\times 3}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $48$ $0$ $940$ $0$ $22400$ $0$ $586404$ $0$ $16248540$
$a_2$ $1$ $3$ $15$ $105$ $963$ $10233$ $117429$ $1409355$ $17432235$ $220488369$ $2838075345$ $37053415215$ $489491331921$
$a_3$ $1$ $0$ $20$ $0$ $2508$ $0$ $555200$ $0$ $146663020$ $0$ $42421800960$ $0$ $12999628539744$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $15$ $8$ $24$ $48$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $20$ $105$ $60$ $204$ $120$ $432$ $940$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $156$ $963$ $552$ $324$ $2052$ $1188$ $4512$ $2600$ $10020$ $22400$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1536$ $10233$ $888$ $5832$ $3348$ $22680$ $12936$ $7408$ $50832$ $28920$ $114480$ $64960$ $258720$ $586404$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $2508$ $16848$ $117429$ $9612$ $66384$ $37656$ $264972$ $21416$ $149616$ $84672$ $600696$ $338520$ $191180$ $1365600$
$$ $768180$ $3111612$ $1747368$ $7104132$ $16248540$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&1&0&2&4&0&2&0&3&0&0&8\\0&4&0&4&0&12&0&0&16&0&8&0&20&32&0\\2&0&10&0&8&0&22&26&0&16&0&42&0&0&88\\0&4&0&8&0&20&0&0&28&0&12&0&44&60&0\\1&0&8&0&15&0&28&30&0&26&0&63&0&0&120\\0&12&0&20&0&76&0&0&108&0&56&0&168&248&0\\2&0&22&0&28&0&78&82&0&64&0&162&0&0&344\\4&0&26&0&30&0&82&108&0&74&0&186&0&0&408\\0&16&0&28&0&108&0&0&172&0&80&0&260&392&0\\2&0&16&0&26&0&64&74&0&70&0&144&0&0&328\\0&8&0&12&0&56&0&0&80&0&48&0&128&192&0\\3&0&42&0&63&0&162&186&0&144&0&375&0&0&792\\0&20&0&44&0&168&0&0&260&0&128&0&428&620&0\\0&32&0&60&0&248&0&0&392&0&192&0&620&948&0\\8&0&88&0&120&0&344&408&0&328&0&792&0&0&1800\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&10&8&15&76&78&108&172&70&48&375&428&948&1800&848&861&1980&1614&436\end{bmatrix}$

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$