Properties

Label 1.6.L.8.3e
  
Name \(L(D_4,D_2)\)
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:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 & 0 & 0 \\0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$L(C_4,C_2)$, $L_1(D_2)$, $L(D_2,C_2)$
Minimal supergroups:$L(J(D_4),D_{4,1})$, $L(O,T)$, $L_2(D_4)$${}^{\times 2}$, $L(J(D_4),J(D_2))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $33$ $0$ $660$ $0$ $17115$ $0$ $491148$ $0$ $14788158$
$a_2$ $1$ $2$ $10$ $71$ $693$ $7912$ $97516$ $1251007$ $16441935$ $219708890$ $2972563650$ $40614086977$ $559386890607$
$a_3$ $1$ $0$ $14$ $0$ $1890$ $0$ $485960$ $0$ $146329554$ $0$ $47048667624$ $0$ $15736416869448$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $10$ $5$ $16$ $33$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $14$ $71$ $41$ $141$ $81$ $300$ $660$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $108$ $693$ $393$ $230$ $1504$ $857$ $3351$ $1890$ $7540$ $17115$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1132$ $7912$ $642$ $4437$ $2502$ $17841$ $9989$ $5620$ $40608$ $22670$ $92890$ $51695$ $213262$ $491148$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1890$ $13298$ $97516$ $7458$ $54191$ $30196$ $223620$ $16850$ $124065$ $68948$ $514843$ $285104$ $158162$ $1188466$
$$ $657111$ $2749327$ $1517922$ $6371400$ $14788158$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&3&0&2&0&2&0&0&6\\0&3&0&2&0&8&0&0&12&0&6&0&13&24&0\\1&0&7&0&5&0&16&17&0&12&0&31&0&0&70\\0&2&0&7&0&14&0&0&19&0&7&0&38&45&0\\1&0&5&0&11&0&18&24&0&23&0&44&0&0&98\\0&8&0&14&0&54&0&0&84&0&40&0&132&202&0\\1&0&16&0&18&0&62&61&0&47&0&127&0&0&294\\3&0&17&0&24&0&61&87&0&68&0&151&0&0&358\\0&12&0&19&0&84&0&0&145&0&67&0&216&345&0\\2&0&12&0&23&0&47&68&0&70&0&124&0&0&302\\0&6&0&7&0&40&0&0&67&0&36&0&98&164&0\\2&0&31&0&44&0&127&151&0&124&0&307&0&0&710\\0&13&0&38&0&132&0&0&216&0&98&0&382&549&0\\0&24&0&45&0&202&0&0&345&0&164&0&549&869&0\\6&0&70&0&98&0&294&358&0&302&0&710&0&0&1712\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&7&7&11&54&62&87&145&70&36&307&382&869&1712&865&869&2094&1774&531\end{bmatrix}$

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$$0$
$a_1=0$$1/4$$0$$0$$0$$0$$0$$0$
$a_3=0$$1/4$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$1/4$$0$$0$$0$$0$$0$$0$