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Properties

Label 1.6.L.48.49a
  
Name \(L_2(J(T))\)
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 \(C_2^2\times A_4\)

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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:$C_2^2\times A_4$
Order:$48$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 \\0 & 0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$L_1(J(T))$, $L_2(T)$, $L_2(J(D_2))$, $L_2(J(C_3))$, $L(J(T),T)$
Minimal supergroups:$L_2(J(O))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $15$ $0$ $205$ $0$ $4025$ $0$ $97587$ $0$ $2683758$
$a_2$ $1$ $2$ $7$ $32$ $212$ $1862$ $19650$ $231044$ $2895114$ $37703636$ $503148512$ $6824520914$ $93633566273$
$a_3$ $1$ $0$ $7$ $0$ $471$ $0$ $90360$ $0$ $25063927$ $0$ $7893946872$ $0$ $2627005198488$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $7$ $3$ $8$ $15$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $7$ $32$ $17$ $50$ $30$ $98$ $205$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $38$ $212$ $119$ $73$ $403$ $240$ $848$ $500$ $1830$ $4025$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $302$ $1862$ $177$ $1063$ $621$ $3935$ $2276$ $1330$ $8640$ $4980$ $19226$ $11025$ $43162$ $97587$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $471$ $2919$ $19650$ $1687$ $11126$ $6360$ $43650$ $3649$ $24738$ $14082$ $98402$ $55579$ $31541$ $223379$
$$ $125783$ $509684$ $286146$ $1167600$ $2683758$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&0&0&0&0&0&1\\0&2&0&1&0&3&0&0&3&0&2&0&2&6&0\\1&0&4&0&1&0&4&6&0&1&0&7&0&0&13\\0&1&0&3&0&4&0&0&4&0&1&0&9&9&0\\0&0&1&0&6&0&5&4&0&6&0&12&0&0&17\\0&3&0&4&0&15&0&0&17&0&10&0&26&40&0\\0&0&4&0&5&0&16&11&0&8&0&26&0&0&53\\2&0&6&0&4&0&11&24&0&11&0&27&0&0&63\\0&3&0&4&0&17&0&0&30&0&12&0&37&62&0\\0&0&1&0&6&0&8&11&0&19&0&21&0&0&53\\0&2&0&1&0&10&0&0&12&0&12&0&18&31&0\\0&0&7&0&12&0&26&27&0&21&0&64&0&0&121\\0&2&0&9&0&26&0&0&37&0&18&0&75&94&0\\0&6&0&9&0&40&0&0&62&0&31&0&94&154&0\\1&0&13&0&17&0&53&63&0&53&0&121&0&0&294\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&3&6&15&16&24&30&19&12&64&75&154&294&157&160&367&317&108\end{bmatrix}$

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$$1/24$$0$$0$$1/3$$1/8$
$a_1=0$$5/16$$1/4$$1/48$$0$$0$$1/6$$1/16$
$a_3=0$$5/16$$1/4$$1/48$$0$$0$$1/6$$1/16$
$a_1=a_3=0$$5/16$$1/4$$1/48$$0$$0$$1/6$$1/16$