Properties

Label 1.6.H.1.1a
  
Name \(H\)
Weight $1$
Degree $6$
Real dimension $3$
Components $1$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)^3\)
Component group \(C_1\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$3$
Components:$1$
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:$u\mapsto\mathrm{diag}(u,\bar u, u, \bar u, u, \bar u)$

Subgroups and supergroups

Maximal subgroups:
Minimal supergroups:$H_{ab}$, $H_{a}$, $H_{s}$, $H_{abc}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $6$ $0$ $90$ $0$ $1860$ $0$ $44730$ $0$ $1172556$ $0$ $32496156$
$a_2$ $1$ $3$ $21$ $183$ $1845$ $20223$ $234129$ $2816523$ $34857909$ $440957055$ $5676091641$ $74106653283$ $978982132401$
$a_3$ $1$ $0$ $32$ $0$ $4920$ $0$ $1109120$ $0$ $293308120$ $0$ $84843343872$ $0$ $25999253294784$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $21$ $12$ $42$ $90$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $32$ $183$ $108$ $390$ $228$ $846$ $1860$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $288$ $1845$ $1068$ $624$ $4050$ $2340$ $8970$ $5160$ $19980$ $44730$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $3000$ $20223$ $1728$ $11556$ $6624$ $45198$ $25764$ $14736$ $101502$ $57720$ $228780$ $129780$ $517230$ $1172556$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $4920$ $33480$ $234129$ $19080$ $132444$ $75096$ $529458$ $42672$ $298908$ $169104$ $1200906$ $676680$ $382080$ $2730660$
$$ $1535940$ $6222594$ $3494232$ $14207508$ $32496156$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&3&0&4&6&0&4&0&9&0&0&16\\0&6&0&6&0&24&0&0&30&0&18&0&42&66&0\\2&0&16&0&18&0&44&48&0&32&0&90&0&0&176\\0&6&0&14&0&40&0&0&54&0&26&0&90&122&0\\3&0&18&0&27&0&54&66&0&48&0&123&0&0&240\\0&24&0&40&0&152&0&0&216&0&112&0&336&496&0\\4&0&44&0&54&0&154&168&0&124&0&324&0&0&688\\6&0&48&0&66&0&168&204&0&156&0&378&0&0&816\\0&30&0&54&0&216&0&0&342&0&162&0&522&786&0\\4&0&32&0&48&0&124&156&0&132&0&288&0&0&656\\0&18&0&26&0&112&0&0&162&0&92&0&252&386&0\\9&0&90&0&123&0&324&378&0&288&0&741&0&0&1584\\0&42&0&90&0&336&0&0&522&0&252&0&852&1242&0\\0&66&0&122&0&496&0&0&786&0&386&0&1242&1886&0\\16&0&176&0&240&0&688&816&0&656&0&1584&0&0&3600\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&6&16&14&27&152&154&204&342&132&92&741&852&1886&3600&1686&1695&3956&3210&862\end{bmatrix}$

Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.