Properties

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

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $27$ $0$ $480$ $0$ $11235$ $0$ $293328$ $0$ $8124732$
$a_2$ $1$ $3$ $12$ $66$ $522$ $5238$ $59079$ $705771$ $8719398$ $110254026$ $1419067197$ $18526796181$ $244745931681$
$a_3$ $1$ $0$ $14$ $0$ $1302$ $0$ $278240$ $0$ $73340470$ $0$ $21211029504$ $0$ $6499816162224$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $12$ $6$ $15$ $27$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $14$ $66$ $36$ $111$ $66$ $225$ $480$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $90$ $522$ $294$ $174$ $1053$ $612$ $2283$ $1320$ $5040$ $11235$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $804$ $5238$ $468$ $2970$ $1710$ $11421$ $6522$ $3744$ $25497$ $14520$ $57330$ $32550$ $129465$ $293328$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1302$ $8532$ $59079$ $4878$ $33354$ $18936$ $132729$ $10788$ $74970$ $42456$ $300591$ $169440$ $95730$ $683070$
$$ $384300$ $1556121$ $873936$ $3552444$ $8124732$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&3&0&1&0&0&0&0&4\\0&3&0&3&0&6&0&0&9&0&3&0&9&15&0\\2&0&7&0&3&0&11&15&0&8&0&18&0&0&44\\0&3&0&5&0&10&0&0&15&0&5&0&21&29&0\\0&0&3&0&9&0&15&12&0&15&0&33&0&0&60\\0&6&0&10&0&38&0&0&54&0&28&0&84&124&0\\1&0&11&0&15&0&40&39&0&34&0&81&0&0&172\\3&0&15&0&12&0&39&60&0&33&0&90&0&0&204\\0&9&0&15&0&54&0&0&87&0&39&0&129&195&0\\1&0&8&0&15&0&34&33&0&39&0&72&0&0&164\\0&3&0&5&0&28&0&0&39&0&26&0&66&95&0\\0&0&18&0&33&0&81&90&0&72&0&192&0&0&396\\0&9&0&21&0&84&0&0&129&0&66&0&216&309&0\\0&15&0&29&0&124&0&0&195&0&95&0&309&479&0\\4&0&44&0&60&0&172&204&0&164&0&396&0&0&900\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&7&5&9&38&40&60&87&39&26&192&216&479&900&429&444&992&816&223\end{bmatrix}$

Event probabilities

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