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Properties

Label 1.6.N.144.125a
  
Name \(J_s(B(T,3))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $144$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_3:\GL(2,3)\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$144$
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:$C_3:\GL(2,3)$
Order:$144$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0\\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 & 0 \\0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 & 0 \\0 &0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}, \begin{bmatrix}\zeta_{9}^{8} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{18}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{18}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{9}^{1} & 0 & 0 \\0 & 0 & 0 & 0 &\zeta_{18}^{17} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{18}^{17} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0\\0 & 0 & 0 & 0 & 0 & -1 \\-i & 0 & 0 & 0 & 0 & 0 \\0 & -i & 0 & 0 & 0 & 0 \\0 &0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$B(T,3)$, $J_s(B(1,12;2))$, $J(A(3,6))$, $J_s(B(T,1))$, $J_s(B(T,1;1))$${}^{\times 2}$
Minimal supergroups:

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $21$ $0$ $340$ $0$ $7245$ $0$ $184212$ $0$ $5214594$
$a_2$ $1$ $2$ $8$ $45$ $349$ $3372$ $37411$ $451411$ $5738851$ $75476586$ $1015970083$ $13906709181$ $192786990336$
$a_3$ $1$ $0$ $8$ $0$ $828$ $0$ $176810$ $0$ $50290268$ $0$ $16143412428$ $0$ $5513234332804$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $8$ $3$ $10$ $21$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $8$ $45$ $23$ $76$ $45$ $158$ $340$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $56$ $349$ $193$ $116$ $693$ $407$ $1495$ $870$ $3270$ $7245$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $516$ $3372$ $300$ $1905$ $1102$ $7258$ $4159$ $2400$ $16104$ $9190$ $36046$ $20475$ $81242$ $184212$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $828$ $5402$ $37411$ $3094$ $21041$ $11941$ $83712$ $6799$ $47148$ $26649$ $189561$ $106407$ $59947$ $431567$
$$ $241528$ $986853$ $550746$ $2264892$ $5214594$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&3&0&0&0&1&0&0&2\\0&2&0&1&0&5&0&0&5&0&4&0&5&11&0\\1&0&5&0&2&0&7&11&0&2&0&13&0&0&26\\0&1&0&4&0&7&0&0&8&0&3&0&16&18&0\\0&0&2&0&9&0&9&7&0&11&0&22&0&0&34\\0&5&0&7&0&26&0&0&33&0&18&0&50&76&0\\0&0&7&0&9&0&28&22&0&19&0&50&0&0&106\\3&0&11&0&7&0&22&43&0&20&0&52&0&0&126\\0&5&0&8&0&33&0&0&56&0&23&0&78&122&0\\0&0&2&0&11&0&19&20&0&30&0&45&0&0&106\\0&4&0&3&0&18&0&0&23&0&18&0&34&60&0\\1&0&13&0&22&0&50&52&0&45&0&117&0&0&242\\0&5&0&16&0&50&0&0&78&0&34&0&138&189&0\\0&11&0&18&0&76&0&0&122&0&60&0&189&300&0\\2&0&26&0&34&0&106&126&0&106&0&242&0&0&589\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&4&9&26&28&43&56&30&18&117&138&300&589&298&318&712&616&186\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$5/9$$0$$1/18$$1/4$$0$$1/4$
$a_1=0$$5/9$$5/9$$0$$1/18$$1/4$$0$$1/4$
$a_3=0$$1/2$$1/2$$0$$0$$1/4$$0$$1/4$
$a_1=a_3=0$$1/2$$1/2$$0$$0$$1/4$$0$$1/4$