Properties

Label 1.6.H.4.2c
  
Name \(H_{a,b}\)
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:$\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0& 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$H_{ab}$, $H_{a}$${}^{\times 2}$
Minimal supergroups:$H_{a,b,c}$${}^{\times 3}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $42$ $0$ $670$ $0$ $13650$ $0$ $324954$ $0$ $8551158$
$a_2$ $1$ $3$ $14$ $84$ $648$ $6048$ $64139$ $737187$ $8915132$ $111482364$ $1426839309$ $18576367857$ $245064424185$
$a_3$ $1$ $0$ $18$ $0$ $1542$ $0$ $290400$ $0$ $73958710$ $0$ $21243414528$ $0$ $6501562803120$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $14$ $8$ $22$ $42$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $18$ $84$ $50$ $156$ $96$ $318$ $670$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $118$ $648$ $384$ $234$ $1332$ $798$ $2854$ $1700$ $6210$ $13650$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $984$ $6048$ $588$ $3528$ $2082$ $13134$ $7664$ $4504$ $29010$ $16860$ $64580$ $37380$ $144550$ $324954$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1542$ $9648$ $64139$ $5622$ $36780$ $21220$ $143272$ $12308$ $81996$ $47136$ $322362$ $183940$ $105390$ $728380$
$$ $414470$ $1651160$ $937188$ $3753162$ $8551158$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&1&0&2&3&0&1&0&2&0&0&4\\0&4&0&4&0&10&0&0&10&0&6&0&12&18&0\\2&0&9&0&7&0&15&17&0&8&0&26&0&0&44\\0&4&0&6&0&14&0&0&16&0&8&0&24&32&0\\1&0&7&0&12&0&19&18&0&13&0&40&0&0&60\\0&10&0&14&0&50&0&0&58&0&36&0&92&132&0\\2&0&15&0&19&0&45&45&0&32&0&90&0&0&172\\3&0&17&0&18&0&45&60&0&35&0&100&0&0&204\\0&10&0&16&0&58&0&0&88&0&42&0&132&198&0\\1&0&8&0&13&0&32&35&0&37&0&70&0&0&164\\0&6&0&8&0&36&0&0&42&0&32&0&72&102&0\\2&0&26&0&40&0&90&100&0&70&0&210&0&0&396\\0&12&0&24&0&92&0&0&132&0&72&0&222&316&0\\0&18&0&32&0&132&0&0&198&0&102&0&316&484&0\\4&0&44&0&60&0&172&204&0&164&0&396&0&0&900\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&9&6&12&50&45&60&88&37&32&210&222&484&900&434&437&998&833&228\end{bmatrix}$

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$