Properties

Label 1.6.H.8.2a
  
Name \(H_{c,at}\)
Weight $1$
Degree $6$
Real dimension $3$
Components $8$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)^3\)
Component group \(C_2\times C_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$3$
Components:$8$
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:$C_2\times C_4$
Order:$8$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 &0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$H_{act}$, $H_{at}$, $H_{a,bc}$
Minimal supergroups:

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $18$ $0$ $290$ $0$ $6230$ $0$ $154602$ $0$ $4169088$
$a_2$ $1$ $2$ $7$ $38$ $293$ $2822$ $30805$ $360740$ $4408633$ $55434098$ $711476627$ $9275791010$ $122452588967$
$a_3$ $1$ $0$ $7$ $0$ $699$ $0$ $142000$ $0$ $36822555$ $0$ $10613578752$ $0$ $3250344268248$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $7$ $3$ $9$ $18$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $7$ $38$ $20$ $65$ $39$ $135$ $290$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $49$ $293$ $165$ $99$ $590$ $348$ $1279$ $750$ $2810$ $6230$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $438$ $2822$ $258$ $1611$ $939$ $6119$ $3533$ $2052$ $13608$ $7830$ $30460$ $17465$ $68495$ $154602$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $699$ $4518$ $30805$ $2607$ $17493$ $10012$ $68940$ $5754$ $39201$ $22368$ $155679$ $88300$ $50245$ $352800$
$$ $199640$ $801759$ $452718$ $1826370$ $4169088$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&0&0&1&0&0&2\\0&2&0&1&0&4&0&0&4&0&3&0&5&9&0\\1&0&4&0&2&0&6&9&0&3&0&11&0&0&22\\0&1&0&3&0&6&0&0&8&0&3&0&12&15&0\\0&0&2&0&7&0&8&6&0&8&0&19&0&0&30\\0&4&0&6&0&22&0&0&28&0&16&0&44&64&0\\0&0&6&0&8&0&23&20&0&17&0&43&0&0&86\\2&0&9&0&6&0&20&33&0&15&0&46&0&0&102\\0&4&0&8&0&28&0&0&44&0&20&0&66&98&0\\0&0&3&0&8&0&17&15&0&20&0&37&0&0&82\\0&3&0&3&0&16&0&0&20&0&15&0&33&50&0\\1&0&11&0&19&0&43&46&0&37&0&101&0&0&198\\0&5&0&12&0&44&0&0&66&0&33&0&110&156&0\\0&9&0&15&0&64&0&0&98&0&50&0&156&240&0\\2&0&22&0&30&0&86&102&0&82&0&198&0&0&450\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&3&7&22&23&33&44&20&15&101&110&240&450&215&228&498&416&112\end{bmatrix}$

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$$1/4$
$a_1=0$$3/8$$3/8$$0$$0$$1/4$$0$$1/8$
$a_3=0$$5/8$$5/8$$0$$0$$1/2$$0$$1/8$
$a_1=a_3=0$$3/8$$3/8$$0$$0$$1/4$$0$$1/8$