Properties

Label 1.6.H.6.2a
  
Name \(H_{abc,s}\)
Weight $1$
Degree $6$
Real dimension $3$
Components $6$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)^3\)
Component group \(C_6\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$3$
Components:$6$
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:$C_6$
Order:$6$
Abelian:yes
Generators:$\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1\\1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $15$ $0$ $310$ $0$ $7455$ $0$ $195426$ $0$ $5416026$
$a_2$ $1$ $1$ $5$ $35$ $321$ $3411$ $39143$ $469785$ $5810745$ $73496123$ $946025115$ $12351138405$ $163163777307$
$a_3$ $1$ $0$ $6$ $0$ $822$ $0$ $184860$ $0$ $48884710$ $0$ $14140557396$ $0$ $4333208882772$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $5$ $2$ $7$ $15$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $6$ $35$ $18$ $65$ $38$ $141$ $310$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $48$ $321$ $178$ $104$ $675$ $390$ $1495$ $860$ $3330$ $7455$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $500$ $3411$ $288$ $1926$ $1104$ $7533$ $4294$ $2456$ $16917$ $9620$ $38130$ $21630$ $86205$ $195426$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $822$ $5580$ $39143$ $3180$ $22074$ $12516$ $88243$ $7112$ $49818$ $28184$ $200151$ $112780$ $63680$ $455110$
$$ $255990$ $1037099$ $582372$ $2367918$ $5416026$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&1&2&0&0&0&1&0&0&2\\0&1&0&1&0&4&0&0&5&0&3&0&7&11&0\\0&0&4&0&2&0&6&10&0&4&0&14&0&0&30\\0&1&0&3&0&6&0&0&9&0&5&0&15&21&0\\0&0&2&0&6&0&10&8&0&10&0&22&0&0&40\\0&4&0&6&0&26&0&0&36&0&18&0&56&82&0\\1&0&6&0&10&0&27&26&0&22&0&55&0&0&114\\2&0&10&0&8&0&26&40&0&22&0&60&0&0&136\\0&5&0&9&0&36&0&0&57&0&27&0&87&131&0\\0&0&4&0&10&0&22&22&0&26&0&50&0&0&108\\0&3&0&5&0&18&0&0&27&0&16&0&42&65&0\\1&0&14&0&22&0&55&60&0&50&0&125&0&0&264\\0&7&0&15&0&56&0&0&87&0&42&0&142&207&0\\0&11&0&21&0&82&0&0&131&0&65&0&207&315&0\\2&0&30&0&40&0&114&136&0&108&0&264&0&0&602\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&4&3&6&26&27&40&57&26&16&125&142&315&602&281&298&660&541&147\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$5/6$$0$$2/3$$0$$0$$1/6$
$a_1=0$$5/6$$5/6$$0$$2/3$$0$$0$$1/6$
$a_3=0$$1/2$$1/2$$0$$1/3$$0$$0$$1/6$
$a_1=a_3=0$$1/2$$1/2$$0$$1/3$$0$$0$$1/6$