Properties

Label 1.6.N.16.13a
  
Name \(J(B(1,4;2)_2)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $16$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(D_4:C_2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$16$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)_3$
$\mathbb{R}$-dimension:$1$
Description:$\left\{\begin{bmatrix}\alpha I_3&0,\\ 0&\bar \alpha I_3\end{bmatrix}: \alpha\bar\alpha=1,\ \alpha\in\mathbb{C}\right\}$ Symplectic form:$\begin{bmatrix} 0 & I_3\\ -I_3 & 0\end{bmatrix}$
Hodge circle:$u\mapsto \mathrm{diag}(u,u, u, \bar u, \bar u, \bar u)$

Component group

Name:$D_4:C_2$
Order:$16$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{7} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0\\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{5} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 & 0 & 0 \\0 & i & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -i \\0 & 0& 0 & 0 & -i & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(A(1,4)_2)$${}^{\times 3}$, $J_s(A(1,4)_2)$${}^{\times 3}$, $B(1,4;2)_2$
Minimal supergroups:$J(B(1,8)_1)$, $J(B(T,1))$, $J(B(2,4))$${}^{\times 2}$, $J(B(1,12;2))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $33$ $0$ $920$ $0$ $28735$ $0$ $930132$ $0$ $30691122$
$a_2$ $1$ $2$ $11$ $89$ $1026$ $13372$ $182084$ $2527387$ $35479502$ $502054424$ $7148206446$ $102280581427$ $1469476480767$
$a_3$ $1$ $0$ $12$ $0$ $2966$ $0$ $966180$ $0$ $335126414$ $0$ $119999282352$ $0$ $43841219257418$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $11$ $4$ $15$ $33$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $12$ $89$ $44$ $176$ $97$ $400$ $920$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $133$ $1026$ $546$ $302$ $2288$ $1254$ $5299$ $2895$ $12320$ $28735$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1714$ $13372$ $939$ $7246$ $3959$ $30918$ $16850$ $9196$ $72207$ $39307$ $168996$ $91889$ $396186$ $930132$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $2966$ $23057$ $182084$ $12575$ $98856$ $53794$ $425991$ $29290$ $231429$ $125804$ $999831$ $542733$ $294784$ $2349792$
$$ $1274595$ $5528649$ $2996868$ $13020546$ $30691122$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&1&5&0&0&0&2&0&0&10\\0&2&0&2&0&9&0&0&16&0&7&0&24&39&0\\1&0&8&0&4&0&19&29&0&20&0&48&0&0&125\\0&2&0&6&0&19&0&0&33&0&14&0&56&84&0\\0&0&4&0&14&0&31&30&0&37&0&74&0&0&180\\0&9&0&19&0&80&0&0&144&0&64&0&234&370&0\\1&0&19&0&31&0&95&107&0&98&0&225&0&0&568\\5&0&29&0&30&0&107&157&0&114&0&271&0&0&711\\0&16&0&33&0&144&0&0&264&0&118&0&427&681&0\\0&0&20&0&37&0&98&114&0&123&0&253&0&0&632\\0&7&0&14&0&64&0&0&118&0&54&0&191&306&0\\2&0&48&0&74&0&225&271&0&253&0&570&0&0&1440\\0&24&0&56&0&234&0&0&427&0&191&0&718&1122&0\\0&39&0&84&0&370&0&0&681&0&306&0&1122&1788&0\\10&0&125&0&180&0&568&711&0&632&0&1440&0&0&3696\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&8&6&14&80&95&157&264&123&54&570&718&1788&3696&1928&2005&4910&4385&1296\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$1/2$$1/8$$0$$0$$0$$3/8$
$a_1=0$$1/2$$1/2$$1/8$$0$$0$$0$$3/8$
$a_3=0$$1/2$$1/2$$1/8$$0$$0$$0$$3/8$
$a_1=a_3=0$$1/2$$1/2$$1/8$$0$$0$$0$$3/8$