Properties

Label 1.6.J.1.1a
  
Name \(J_1(E_1)\)
Weight $1$
Degree $6$
Real dimension $4$
Components $1$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{SU}(2)_2\)
Component group \(C_1\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$4$
Components:$1$
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:$\mathrm{diag}(u,\bar u, u,\bar u,\bar u,u)$

Subgroups and supergroups

Maximal subgroups:
Minimal supergroups:$J_2(E_1)$, $J_1(E_2)$, $J_1(E_3)$, $J(J(E_1),E_1)$, $J(E_2,E_1)$, $J_1(J(E_1))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $6$ $0$ $86$ $0$ $1660$ $0$ $37254$ $0$ $916020$ $0$ $23952060$
$a_2$ $1$ $4$ $25$ $197$ $1811$ $18370$ $199015$ $2258800$ $26541115$ $320350484$ $3950530883$ $49581281117$ $631461728005$
$a_3$ $1$ $0$ $36$ $0$ $4676$ $0$ $913040$ $0$ $215523812$ $0$ $56803010448$ $0$ $16090127983248$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $4$ $6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $25$ $14$ $44$ $86$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $36$ $197$ $114$ $386$ $226$ $792$ $1660$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $302$ $1811$ $1048$ $612$ $3762$ $2182$ $7990$ $4620$ $17176$ $37254$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $2892$ $18370$ $1678$ $10564$ $6098$ $39466$ $22664$ $13048$ $85774$ $49124$ $187836$ $107282$ $413784$ $916020$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $4676$ $30010$ $199015$ $17250$ $113574$ $64964$ $436600$ $37236$ $248636$ $141852$ $964290$ $547856$ $311788$ $2140524$
$$ $1213422$ $4771350$ $2699172$ $10673640$ $23952060$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&3&0&2&0&5&7&0&4&0&7&0&0&16\\0&6&0&8&0&24&0&0&32&0&12&0&38&60&0\\3&0&18&0&17&0&44&51&0&33&0&74&0&0&160\\0&8&0&14&0&40&0&0&56&0&22&0&74&110&0\\2&0&17&0&23&0&53&58&0&45&0&101&0&0&208\\0&24&0&40&0&140&0&0&200&0&88&0&280&416&0\\5&0&44&0&53&0&140&155&0&117&0&268&0&0&576\\7&0&51&0&58&0&155&186&0&133&0&306&0&0&672\\0&32&0&56&0&200&0&0&302&0&132&0&430&644&0\\4&0&33&0&45&0&117&133&0&110&0&240&0&0&528\\0&12&0&22&0&88&0&0&132&0&68&0&202&292&0\\7&0&74&0&101&0&268&306&0&240&0&571&0&0&1232\\0&38&0&74&0&280&0&0&430&0&202&0&656&960&0\\0&60&0&110&0&416&0&0&644&0&292&0&960&1450&0\\16&0&160&0&208&0&576&672&0&528&0&1232&0&0&2752\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&6&18&14&23&140&140&186&302&110&68&571&656&1450&2752&1262&1269&2880&2288&590\end{bmatrix}$

Event probabilities

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