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Properties

Label 1.6.N.96.193a
  
Name \(J(B(O,1))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $96$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(\GL(2,3):C_2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$96$
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:$\GL(2,3):C_2$
Order:$96$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0\\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 & 0 \\0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 & 0 \\0 &0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{24}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{24}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 &\zeta_{24}^{23} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{24}^{23} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J_s(B(3,2))$, $J(B(1,8)_1)$, $J(B(T,1))$, $J_s(B(T,1))$, $B(O,1)$
Minimal supergroups:

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $21$ $0$ $330$ $0$ $7000$ $0$ $184212$ $0$ $5514894$
$a_2$ $1$ $2$ $8$ $44$ $337$ $3277$ $37376$ $470080$ $6260932$ $86193536$ $1209847303$ $17183876635$ $245939269043$
$a_3$ $1$ $0$ $8$ $0$ $815$ $0$ $183520$ $0$ $57615957$ $0$ $20147153238$ $0$ $7319726711000$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $8$ $3$ $10$ $21$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $8$ $44$ $23$ $75$ $45$ $155$ $330$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $55$ $337$ $189$ $116$ $674$ $400$ $1454$ $850$ $3170$ $7000$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $503$ $3277$ $294$ $1858$ $1079$ $7112$ $4073$ $2358$ $15861$ $9027$ $35672$ $20153$ $80794$ $184212$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $815$ $5311$ $37376$ $3044$ $20917$ $11824$ $84492$ $6711$ $47244$ $26543$ $193069$ $107443$ $60091$ $443553$
$$ $245826$ $1023715$ $565194$ $2371992$ $5514894$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&3&0&0&0&1&0&0&2\\0&2&0&1&0&5&0&0&5&0&4&0&5&10&0\\1&0&5&0&2&0&7&10&0&2&0&13&0&0&25\\0&1&0&4&0&7&0&0&7&0&3&0&17&16&0\\0&0&2&0&9&0&9&7&0&12&0&20&0&0&34\\0&5&0&7&0&25&0&0&32&0&17&0&49&72&0\\0&0&7&0&9&0&28&20&0&16&0&49&0&0&105\\3&0&10&0&7&0&20&41&0&23&0&51&0&0&128\\0&5&0&7&0&32&0&0&55&0&25&0&75&125&0\\0&0&2&0&12&0&16&23&0&35&0&46&0&0&112\\0&4&0&3&0&17&0&0&25&0&16&0&32&59&0\\1&0&13&0&20&0&49&51&0&46&0&112&0&0&252\\0&5&0&17&0&49&0&0&75&0&32&0&145&192&0\\0&10&0&16&0&72&0&0&125&0&59&0&192&313&0\\2&0&25&0&34&0&105&128&0&112&0&252&0&0&637\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&4&9&25&28&41&55&35&16&112&145&313&637&351&353&856&761&264\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$7/12$$1/48$$1/12$$1/8$$1/6$$3/16$
$a_1=0$$7/12$$7/12$$1/48$$1/12$$1/8$$1/6$$3/16$
$a_3=0$$1/2$$1/2$$1/48$$0$$1/8$$1/6$$3/16$
$a_1=a_3=0$$1/2$$1/2$$1/48$$0$$1/8$$1/6$$3/16$