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Properties

Label 1.6.L.48.48f
  
Name \(L_2(O)\)
Weight $1$
Degree $6$
Real dimension $6$
Components $48$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{U}(1)_2\)
Component group \(C_2\times S_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$6$
Components:$48$
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:$C_2\times S_4$
Order:$48$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 \\0 & 0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0& 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $27$ $0$ $380$ $0$ $6965$ $0$ $151578$ $0$ $3729264$
$a_2$ $1$ $2$ $9$ $51$ $370$ $3192$ $31275$ $336205$ $3871530$ $46943796$ $591831379$ $7684868655$ $102051564840$
$a_3$ $1$ $0$ $12$ $0$ $855$ $0$ $134650$ $0$ $31177055$ $0$ $8806736862$ $0$ $2769255052674$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $9$ $5$ $14$ $27$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $12$ $51$ $31$ $93$ $57$ $184$ $380$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $71$ $370$ $222$ $138$ $737$ $447$ $1531$ $920$ $3240$ $6965$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $553$ $3192$ $333$ $1889$ $1129$ $6718$ $3974$ $2370$ $14427$ $8500$ $31340$ $18375$ $68670$ $151578$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $855$ $4993$ $31275$ $2954$ $18192$ $10647$ $68078$ $6256$ $39489$ $23008$ $149778$ $86550$ $50250$ $331920$
$$ $191135$ $739893$ $424620$ $1657488$ $3729264$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&2&0&1&0&1&0&0&2\\0&3&0&2&0&6&0&0&6&0&4&0&5&10&0\\1&0&6&0&4&0&9&9&0&4&0&14&0&0&22\\0&2&0&5&0&8&0&0&8&0&3&0&15&15&0\\1&0&4&0&8&0&9&11&0&8&0&20&0&0&28\\0&6&0&8&0&28&0&0&30&0&18&0&44&64&0\\1&0&9&0&9&0&27&22&0&13&0&43&0&0&80\\2&0&9&0&11&0&22&33&0&19&0&47&0&0&92\\0&6&0&8&0&30&0&0&46&0&20&0&58&90&0\\1&0&4&0&8&0&13&19&0&24&0&30&0&0&72\\0&4&0&3&0&18&0&0&20&0&18&0&29&48&0\\1&0&14&0&20&0&43&47&0&30&0&98&0&0&170\\0&5&0&15&0&44&0&0&58&0&29&0&107&134&0\\0&10&0&15&0&64&0&0&90&0&48&0&134&211&0\\2&0&22&0&28&0&80&92&0&72&0&170&0&0&380\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&6&5&8&28&27&33&46&24&18&98&107&211&380&191&185&422&352&116\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$0$$0$$0$$0$$0$$0$
$a_1=0$$3/16$$0$$0$$0$$0$$0$$0$
$a_3=0$$3/16$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$3/16$$0$$0$$0$$0$$0$$0$