Properties

Label 1.6.L.96.226a
  
Name \(L_2(J(O))\)
Weight $1$
Degree $6$
Real dimension $6$
Components $96$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{U}(1)_2\)
Component group \(C_2^2\times S_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$6$
Components:$96$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)\times\mathrm{U}(1)_2$
$\mathbb{R}$-dimension:$6$
Description:$\left\{\begin{bmatrix}A&0&0\\0&\alpha I_2&0\\0&0&\bar\alpha I_2\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),\ \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ 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:$u\mapsto\mathrm{diag}(u,\bar u, u, u,\bar u, \bar u)$

Component group

Name:$C_2^2\times S_4$
Order:$96$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 \\0 & 0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0& 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0& -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0& 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$L_1(J(O))$, $L_2(O)$, $L(J(O),J(T))$, $L_2(J(D_3))$, $L(J(O),O_1)$, $L_2(J(T))$, $L(J(O),O)$, $L_2(O_1)$, $L_2(J(D_4))$
Minimal supergroups:

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $15$ $0$ $195$ $0$ $3500$ $0$ $75852$ $0$ $1864863$
$a_2$ $1$ $2$ $7$ $32$ $203$ $1647$ $15785$ $168534$ $1937038$ $23475674$ $295926932$ $3842467884$ $51025882748$
$a_3$ $1$ $0$ $7$ $0$ $438$ $0$ $67460$ $0$ $15590400$ $0$ $4403395332$ $0$ $1384627920654$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $7$ $3$ $8$ $15$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $7$ $32$ $17$ $49$ $30$ $95$ $195$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $38$ $203$ $115$ $72$ $375$ $228$ $773$ $465$ $1630$ $3500$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $284$ $1647$ $171$ $956$ $572$ $3377$ $1999$ $1195$ $7233$ $4265$ $15695$ $9205$ $34370$ $75852$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $438$ $2518$ $15785$ $1492$ $9129$ $5346$ $34090$ $3143$ $19779$ $11529$ $74943$ $43315$ $25160$ $166025$
$$ $95620$ $370034$ $212373$ $828870$ $1864863$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&0&0&0&0&0&1\\0&2&0&1&0&3&0&0&3&0&2&0&2&5&0\\1&0&4&0&1&0&4&6&0&1&0&6&0&0&11\\0&1&0&3&0&4&0&0&4&0&1&0&8&7&0\\0&0&1&0&6&0&5&3&0&6&0&11&0&0&14\\0&3&0&4&0&14&0&0&15&0&9&0&22&32&0\\0&0&4&0&5&0&15&9&0&7&0&22&0&0&40\\2&0&6&0&3&0&9&22&0&7&0&21&0&0&46\\0&3&0&4&0&15&0&0&24&0&10&0&28&45&0\\0&0&1&0&6&0&7&7&0&16&0&16&0&0&36\\0&2&0&1&0&9&0&0&10&0&10&0&14&24&0\\0&0&6&0&11&0&22&21&0&16&0&51&0&0&85\\0&2&0&8&0&22&0&0&28&0&14&0&56&66&0\\0&5&0&7&0&32&0&0&45&0&24&0&66&107&0\\1&0&11&0&14&0&40&46&0&36&0&85&0&0&190\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&3&6&14&15&22&24&16&10&51&56&107&190&99&104&217&185&66\end{bmatrix}$

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$$1/48$$0$$1/8$$1/6$$3/16$
$a_1=0$$11/32$$1/4$$1/96$$0$$1/16$$1/12$$3/32$
$a_3=0$$13/32$$5/16$$1/96$$0$$1/8$$1/12$$3/32$
$a_1=a_3=0$$11/32$$1/4$$1/96$$0$$1/16$$1/12$$3/32$