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Properties

Label 1.6.N.96.201a
  
Name \(J(B(T,2))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $96$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(\SL(2,3):C_2^2\)

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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:$\SL(2,3):C_2^2$
Order:$96$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0\\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 & 0 \\0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 & 0 \\0 &0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{5} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{1} & 0 & 0 \\0 & 0 & 0 & 0 &\zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{11} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J_n(A(3,4))$, $J(B(T,1))$${}^{\times 2}$, $J(B(2,4))$, $B(T,2)$
Minimal supergroups:$J(B(O,2))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $21$ $0$ $340$ $0$ $7385$ $0$ $194292$ $0$ $5741274$
$a_2$ $1$ $2$ $8$ $44$ $346$ $3432$ $39291$ $490800$ $6470638$ $88237340$ $1229321533$ $17366888640$ $247644695513$
$a_3$ $1$ $0$ $8$ $0$ $844$ $0$ $191450$ $0$ $58922108$ $0$ $20334577188$ $0$ $7345419434122$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $8$ $3$ $10$ $21$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $8$ $44$ $23$ $76$ $45$ $158$ $340$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $56$ $346$ $194$ $117$ $700$ $411$ $1516$ $880$ $3325$ $7385$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $522$ $3432$ $303$ $1944$ $1122$ $7482$ $4270$ $2458$ $16704$ $9486$ $37604$ $21231$ $85218$ $194292$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $844$ $5578$ $39291$ $3183$ $22005$ $12421$ $88792$ $7038$ $49672$ $27904$ $202593$ $112888$ $63174$ $464659$
$$ $257999$ $1070377$ $592326$ $2474892$ $5741274$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&3&0&0&0&1&0&0&2\\0&2&0&1&0&5&0&0&5&0&4&0&5&11&0\\1&0&5&0&2&0&7&10&0&3&0&14&0&0&27\\0&1&0&4&0&7&0&0&8&0&3&0&17&18&0\\0&0&2&0&9&0&9&8&0&11&0&21&0&0&36\\0&5&0&7&0&26&0&0&34&0&18&0&52&78&0\\0&0&7&0&9&0&29&23&0&18&0&51&0&0&112\\3&0&10&0&8&0&23&41&0&24&0&56&0&0&135\\0&5&0&8&0&34&0&0&58&0&25&0&82&131&0\\0&0&3&0&11&0&18&24&0&32&0&48&0&0&116\\0&4&0&3&0&18&0&0&25&0&18&0&35&63&0\\1&0&14&0&21&0&51&56&0&48&0&120&0&0&264\\0&5&0&17&0&52&0&0&82&0&35&0&150&203&0\\0&11&0&18&0&78&0&0&131&0&63&0&203&328&0\\2&0&27&0&36&0&112&135&0&116&0&264&0&0&658\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&4&9&26&29&41&58&32&18&120&150&328&658&350&356&856&762&246\end{bmatrix}$

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$$1/24$$1/12$$0$$1/3$$1/8$
$a_1=0$$7/12$$7/12$$1/24$$1/12$$0$$1/3$$1/8$
$a_3=0$$1/2$$1/2$$1/24$$0$$0$$1/3$$1/8$
$a_1=a_3=0$$1/2$$1/2$$1/24$$0$$0$$1/3$$1/8$