Properties

Label 1.6.J.4.2e
  
Name \(J_2(J(E_1))\)
Weight $1$
Degree $6$
Real dimension $4$
Components $4$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{SU}(2)_2\)
Component group \(C_2^2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$4$
Components:$4$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)\times\mathrm{SU}(2)_2$
$\mathbb{R}$-dimension:$4$
Description:$\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&\overline{B}\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),B\in\mathrm{SU}(2)\right\}$ Symplectic form:$\begin{bmatrix}J_2&0&0\\0&0&I_2\\0&-I_2&0\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$
Hodge circle:$\mathrm{diag}(u,\bar u, u,\bar u,\bar u,u)$

Component group

Name:$C_2^2$
Order:$4$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J_2(E_1)$, $J(J(E_1),E_1)$, $J_1(J(E_1))$
Minimal supergroups:$J_2(J(E_2))$${}^{\times 2}$, $J_2(J(E_3))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $31$ $0$ $500$ $0$ $10227$ $0$ $239820$ $0$ $6123414$
$a_2$ $1$ $3$ $13$ $75$ $563$ $5113$ $52411$ $579085$ $6716539$ $80561025$ $990454079$ $12412427833$ $157970582341$
$a_3$ $1$ $0$ $14$ $0$ $1306$ $0$ $233480$ $0$ $54111554$ $0$ $14211778536$ $0$ $4023085857384$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $13$ $6$ $17$ $31$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $14$ $75$ $40$ $125$ $74$ $245$ $500$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $98$ $563$ $318$ $188$ $1087$ $638$ $2253$ $1320$ $4768$ $10227$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $834$ $5113$ $488$ $2932$ $1708$ $10661$ $6174$ $3592$ $22891$ $13224$ $49720$ $28630$ $108857$ $239820$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1306$ $8082$ $52411$ $4678$ $29978$ $17254$ $113623$ $9964$ $65048$ $37344$ $249501$ $142476$ $81566$ $551564$
$$ $314160$ $1225565$ $696360$ $2734584$ $6123414$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&4&0&0&0&1&0&0&4\\0&3&0&3&0&8&0&0&9&0&3&0&9&17&0\\2&0&8&0&4&0&12&18&0&6&0&18&0&0&40\\0&3&0&5&0&12&0&0&15&0&5&0&19&29&0\\0&0&4&0&10&0&16&12&0&14&0&30&0&0&52\\0&8&0&12&0&40&0&0&52&0&24&0&72&108&0\\1&0&12&0&16&0&39&38&0&30&0&71&0&0&144\\4&0&18&0&12&0&38&58&0&28&0&74&0&0&168\\0&9&0&15&0&52&0&0&77&0&33&0&107&163&0\\0&0&6&0&14&0&30&28&0&32&0&62&0&0&132\\0&3&0&5&0&24&0&0&33&0&22&0&54&73&0\\1&0&18&0&30&0&71&74&0&62&0&153&0&0&308\\0&9&0&19&0&72&0&0&107&0&54&0&170&239&0\\0&17&0&29&0&108&0&0&163&0&73&0&239&369&0\\4&0&40&0&52&0&144&168&0&132&0&308&0&0&688\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&8&5&10&40&39&58&77&32&22&153&170&369&688&323&340&726&589&155\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$0$$0$$0$$0$$0$$0$
$a_1=0$$1/4$$0$$0$$0$$0$$0$$0$
$a_3=0$$1/4$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$1/4$$0$$0$$0$$0$$0$$0$