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Properties

Label 1.6.L.8.3j
  
Name \(L_1(D_{4,2})\)
Weight $1$
Degree $6$
Real dimension $6$
Components $8$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{U}(1)_2\)
Component group \(D_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$6$
Components:$8$
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:$D_4$
Order:$8$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$L_1(D_{2,1})$${}^{\times 2}$, $L_1(C_4)$
Minimal supergroups:$L_1(J(D_4))$, $L(J(D_4),D_{4,2})$, $L_2(D_{4,2})$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $48$ $0$ $940$ $0$ $22470$ $0$ $595224$ $0$ $16850064$
$a_2$ $1$ $3$ $15$ $105$ $964$ $10288$ $119220$ $1453420$ $18355494$ $237997122$ $3148996960$ $42330272020$ $576201094633$
$a_3$ $1$ $0$ $20$ $0$ $2514$ $0$ $570080$ $0$ $158421970$ $0$ $48870259200$ $0$ $16020797563848$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $15$ $8$ $24$ $48$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $20$ $105$ $60$ $204$ $120$ $432$ $940$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $156$ $964$ $552$ $324$ $2054$ $1188$ $4518$ $2600$ $10040$ $22470$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1538$ $10288$ $888$ $5850$ $3352$ $22822$ $12984$ $7420$ $51222$ $29060$ $115580$ $65380$ $261842$ $595224$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $2514$ $16966$ $119220$ $9654$ $67148$ $37962$ $269634$ $21524$ $151638$ $85500$ $612998$ $343960$ $193480$ $1398228$
$$ $782908$ $3198146$ $1787184$ $7332864$ $16850064$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&1&0&2&4&0&2&0&3&0&0&8\\0&4&0&4&0&12&0&0&16&0&8&0&20&32&0\\2&0&10&0&8&0&22&26&0&16&0&42&0&0&88\\0&4&0&8&0&20&0&0&28&0&12&0&44&60&0\\1&0&8&0&15&0&28&30&0&26&0&63&0&0&120\\0&12&0&20&0&76&0&0&108&0&56&0&168&250&0\\2&0&22&0&28&0&78&82&0&64&0&162&0&0&348\\4&0&26&0&30&0&82&109&0&75&0&187&0&0&416\\0&16&0&28&0&108&0&0&174&0&80&0&262&400&0\\2&0&16&0&26&0&64&75&0&73&0&145&0&0&340\\0&8&0&12&0&56&0&0&80&0&48&0&128&194&0\\3&0&42&0&63&0&162&187&0&145&0&378&0&0&808\\0&20&0&44&0&168&0&0&262&0&128&0&434&634&0\\0&32&0&60&0&250&0&0&400&0&194&0&634&980&0\\8&0&88&0&120&0&348&416&0&340&0&808&0&0&1884\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&10&8&15&76&78&109&174&73&48&378&434&980&1884&916&932&2186&1838&522\end{bmatrix}$

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$$0$$0$$0$$0$$1/2$
$a_1=0$$0$$0$$0$$0$$0$$0$$0$
$a_3=0$$0$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$0$$0$$0$$0$$0$$0$$0$