Properties

Label 1.6.H.3.1a
  
Name \(H_{s}\)
Weight $1$
Degree $6$
Real dimension $3$
Components $3$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)^3\)
Component group \(C_3\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$3$
Components:$3$
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:$C_3$
Order:$3$
Abelian:yes
Generators:$\begin{bmatrix}0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 &0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $30$ $0$ $620$ $0$ $14910$ $0$ $390852$ $0$ $10832052$
$a_2$ $1$ $1$ $7$ $61$ $615$ $6741$ $78043$ $938841$ $11619303$ $146985685$ $1892030547$ $24702217761$ $326327377467$
$a_3$ $1$ $0$ $12$ $0$ $1644$ $0$ $369720$ $0$ $97769420$ $0$ $28281114792$ $0$ $8666417765544$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $7$ $4$ $14$ $30$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $12$ $61$ $36$ $130$ $76$ $282$ $620$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $96$ $615$ $356$ $208$ $1350$ $780$ $2990$ $1720$ $6660$ $14910$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1000$ $6741$ $576$ $3852$ $2208$ $15066$ $8588$ $4912$ $33834$ $19240$ $76260$ $43260$ $172410$ $390852$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1644$ $11160$ $78043$ $6360$ $44148$ $25032$ $176486$ $14224$ $99636$ $56368$ $400302$ $225560$ $127360$ $910220$
$$ $511980$ $2074198$ $1164744$ $4735836$ $10832052$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&1&0&2&2&0&2&0&3&0&0&4\\0&2&0&2&0&8&0&0&10&0&6&0&14&22&0\\0&0&6&0&6&0&14&16&0&10&0&30&0&0&60\\0&2&0&6&0&12&0&0&18&0&10&0&30&42&0\\1&0&6&0&9&0&18&22&0&16&0&41&0&0&80\\0&8&0&12&0&52&0&0&72&0&36&0&112&164&0\\2&0&14&0&18&0&52&56&0&42&0&108&0&0&228\\2&0&16&0&22&0&56&68&0&52&0&126&0&0&272\\0&10&0&18&0&72&0&0&114&0&54&0&174&262&0\\2&0&10&0&16&0&42&52&0&46&0&96&0&0&216\\0&6&0&10&0&36&0&0&54&0&32&0&84&130&0\\3&0&30&0&41&0&108&126&0&96&0&247&0&0&528\\0&14&0&30&0&112&0&0&174&0&84&0&284&414&0\\0&22&0&42&0&164&0&0&262&0&130&0&414&630&0\\4&0&60&0&80&0&228&272&0&216&0&528&0&0&1204\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&6&9&52&52&68&114&46&32&247&284&630&1204&562&569&1320&1070&294\end{bmatrix}$

Event probabilities

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