Properties

Label 1.6.E.6.1a
  
Name \(E_{s,t}\)
Weight $1$
Degree $6$
Real dimension $9$
Components $6$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{SU}(2)^3\)
Component group \(S_3\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$9$
Components:$6$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{SU}(2)^3$
$\mathbb{R}$-dimension:$9$
Description:$\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}: A,B,C\in\mathrm{SU}(2)\right\}$ 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:$S_3$
Order:$6$
Abelian:no
Generators:$\begin{bmatrix}0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 &0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$E_{s}$, $E_{t}$
Minimal supergroups:

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $5$ $0$ $50$ $0$ $714$ $0$ $12222$ $0$ $233904$
$a_2$ $1$ $1$ $3$ $12$ $65$ $436$ $3377$ $28792$ $262817$ $2526496$ $25309505$ $262280932$ $2796539990$
$a_3$ $1$ $0$ $3$ $0$ $124$ $0$ $12275$ $0$ $1705536$ $0$ $289716588$ $0$ $56419337172$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $3$ $1$ $3$ $5$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $3$ $12$ $6$ $16$ $9$ $27$ $50$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $13$ $65$ $36$ $22$ $105$ $63$ $192$ $115$ $365$ $714$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $85$ $436$ $50$ $254$ $152$ $791$ $474$ $287$ $1523$ $915$ $3004$ $1799$ $6020$ $12222$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $124$ $628$ $3377$ $376$ $1997$ $1194$ $6542$ $715$ $3902$ $2334$ $13058$ $7782$ $4652$ $26468$
$$ $15743$ $54257$ $32186$ $112224$ $233904$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&0&1&0&0&0&0&0&0&0\\0&1&0&0&0&1&0&0&1&0&0&0&0&2&0\\0&0&2&0&0&0&1&2&0&0&0&1&0&0&3\\0&0&0&2&0&1&0&0&1&0&0&0&2&2&0\\0&0&0&0&2&0&1&1&0&2&0&2&0&0&3\\0&1&0&1&0&4&0&0&4&0&1&0&4&6&0\\0&0&1&0&1&0&5&2&0&1&0&4&0&0&7\\1&0&2&0&1&0&2&6&0&2&0&3&0&0&8\\0&1&0&1&0&4&0&0&6&0&2&0&4&8&0\\0&0&0&0&2&0&1&2&0&4&0&3&0&0&5\\0&0&0&0&0&1&0&0&2&0&3&0&2&3&0\\0&0&1&0&2&0&4&3&0&3&0&8&0&0&12\\0&0&0&2&0&4&0&0&4&0&2&0&10&8&0\\0&2&0&2&0&6&0&0&8&0&3&0&8&16&0\\0&0&3&0&3&0&7&8&0&5&0&12&0&0&24\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&2&2&4&5&6&6&4&3&8&10&16&24&12&15&23&15&7\end{bmatrix}$

Event probabilities

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