Properties

Label 1.6.H.4.1b
  
Name \(H_{at}\)
Weight $1$
Degree $6$
Real dimension $3$
Components $4$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)^3\)
Component group \(C_4\)

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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:$C_4$
Order:$4$
Abelian:yes
Generators:$\begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$H_{ab}$
Minimal supergroups:$H_{c,at}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $27$ $0$ $480$ $0$ $11235$ $0$ $293328$ $0$ $8124732$
$a_2$ $1$ $2$ $8$ $53$ $482$ $5117$ $58715$ $704678$ $8716118$ $110244185$ $1419037673$ $18526707608$ $244745665961$
$a_3$ $1$ $0$ $10$ $0$ $1254$ $0$ $277600$ $0$ $73331510$ $0$ $21210900480$ $0$ $6499814269872$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $8$ $4$ $13$ $27$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $10$ $53$ $30$ $103$ $60$ $219$ $480$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $78$ $482$ $276$ $162$ $1027$ $594$ $2259$ $1300$ $5020$ $11235$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $768$ $5117$ $444$ $2916$ $1674$ $11341$ $6468$ $3704$ $25419$ $14460$ $57250$ $32480$ $129395$ $293328$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1254$ $8424$ $58715$ $4806$ $33192$ $18828$ $132487$ $10708$ $74808$ $42336$ $300351$ $169260$ $95590$ $682810$
$$ $384090$ $1555841$ $873684$ $3552192$ $8124732$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&0&2&0&1&0&2&0&0&4\\0&3&0&1&0&6&0&0&7&0&5&0&9&17&0\\1&0&5&0&4&0&11&13&0&8&0&21&0&0&44\\0&1&0&5&0&10&0&0&15&0&5&0&23&29&0\\1&0&4&0&9&0&12&15&0&13&0&32&0&0&60\\0&6&0&10&0&38&0&0&54&0&28&0&84&124&0\\0&0&11&0&12&0&42&41&0&32&0&80&0&0&172\\2&0&13&0&15&0&41&54&0&37&0&93&0&0&204\\0&7&0&15&0&54&0&0&87&0&39&0&131&195&0\\1&0&8&0&13&0&32&37&0&35&0&72&0&0&164\\0&5&0&5&0&28&0&0&39&0&26&0&62&97&0\\2&0&21&0&32&0&80&93&0&72&0&188&0&0&396\\0&9&0&23&0&84&0&0&131&0&62&0&216&309&0\\0&17&0&29&0&124&0&0&195&0&97&0&309&475&0\\4&0&44&0&60&0&172&204&0&164&0&396&0&0&900\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&5&5&9&38&42&54&87&35&26&188&216&475&900&425&432&992&810&219\end{bmatrix}$

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