Properties

Label 1.6.H.4.2b
  
Name \(H_{a,bc}\)
Weight $1$
Degree $6$
Real dimension $3$
Components $4$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)^3\)
Component group \(C_2^2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$3$
Components:$4$
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:$C_2^2$
Order:$4$
Abelian:yes
Generators:$\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0& 0 & 0 & -1 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$H_{ab}$, $H_{a}$, $H_{abc}$
Minimal supergroups:$H_{c,at}$, $H_{a,b,c}$${}^{\times 3}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $33$ $0$ $570$ $0$ $12425$ $0$ $309078$ $0$ $8337714$
$a_2$ $1$ $3$ $13$ $75$ $585$ $5643$ $61609$ $721479$ $8817265$ $110868195$ $1422953253$ $18551582019$ $244905177933$
$a_3$ $1$ $0$ $14$ $0$ $1398$ $0$ $284000$ $0$ $73645110$ $0$ $21227157504$ $0$ $6500688536496$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $13$ $6$ $17$ $33$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $14$ $75$ $40$ $129$ $78$ $267$ $570$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $98$ $585$ $330$ $198$ $1179$ $696$ $2555$ $1500$ $5610$ $12425$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $876$ $5643$ $516$ $3222$ $1878$ $12237$ $7066$ $4104$ $27213$ $15660$ $60910$ $34930$ $136955$ $309078$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1398$ $9036$ $61609$ $5214$ $34986$ $20024$ $137879$ $11508$ $78402$ $44736$ $311355$ $176600$ $100490$ $705590$
$$ $399280$ $1603483$ $905436$ $3652614$ $8337714$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&4&0&0&0&1&0&0&4\\0&3&0&3&0&8&0&0&9&0&5&0&11&17&0\\2&0&8&0&4&0&12&18&0&6&0&22&0&0&44\\0&3&0&5&0&12&0&0&15&0&7&0&23&31&0\\0&0&4&0&12&0&18&12&0&16&0&38&0&0&60\\0&8&0&12&0&44&0&0&56&0&32&0&88&128&0\\1&0&12&0&18&0&43&40&0&34&0&87&0&0&172\\4&0&18&0&12&0&40&66&0&30&0&92&0&0&204\\0&9&0&15&0&56&0&0&87&0&41&0&131&197&0\\0&0&6&0&16&0&34&30&0&40&0&74&0&0&164\\0&5&0&7&0&32&0&0&41&0&28&0&68&99&0\\1&0&22&0&38&0&87&92&0&74&0&201&0&0&396\\0&11&0&23&0&88&0&0&131&0&68&0&218&313&0\\0&17&0&31&0&128&0&0&197&0&99&0&313&479&0\\4&0&44&0&60&0&172&204&0&164&0&396&0&0&900\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&8&5&12&44&43&66&87&40&28&201&218&479&900&429&454&994&829&223\end{bmatrix}$

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$$1/2$
$a_1=0$$1/4$$1/4$$0$$0$$0$$0$$1/4$
$a_3=0$$1/4$$1/4$$0$$0$$0$$0$$1/4$
$a_1=a_3=0$$1/4$$1/4$$0$$0$$0$$0$$1/4$