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Properties

Label 1.6.L.8.5b
  
Name \(L(J(D_2),J(C_2))\)
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 \(C_2^3\)

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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:$C_2^3$
Order:$8$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 &0 & 0 \\0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 &0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$L(D_{2,1},C_{2,1})$${}^{\times 2}$, $L_1(J(C_2))$, $L(D_2,C_2)$, $L(J(C_2),J(C_1))$${}^{\times 2}$, $L(D_{2,1},C_2)$
Minimal supergroups:$L(J(D_6),J(D_3))$, $L(J(D_4),J(C_4))$${}^{\times 2}$, $L_2(J(D_2))$${}^{\times 3}$, $L(J(D_4),J(D_2))$, $L(J(D_6),J(C_6))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $27$ $0$ $620$ $0$ $16835$ $0$ $489132$ $0$ $14773374$
$a_2$ $1$ $2$ $10$ $68$ $670$ $7772$ $96757$ $1247108$ $16422490$ $219613460$ $2972099335$ $40611837488$ $559376011135$
$a_3$ $1$ $0$ $11$ $0$ $1809$ $0$ $483620$ $0$ $146261129$ $0$ $47046651876$ $0$ $15736357046916$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $10$ $3$ $12$ $27$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $11$ $68$ $34$ $125$ $69$ $274$ $620$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $93$ $670$ $361$ $209$ $1432$ $808$ $3238$ $1810$ $7360$ $16835$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1070$ $7772$ $594$ $4295$ $2401$ $17513$ $9763$ $5470$ $40089$ $22320$ $92076$ $51135$ $211974$ $489132$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1809$ $13014$ $96757$ $7265$ $53537$ $29750$ $222097$ $16530$ $123033$ $68238$ $512438$ $283476$ $157077$ $1184676$
$$ $654570$ $2743370$ $1513890$ $6361992$ $14773374$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&5&0&1&0&1&0&0&6\\0&2&0&1&0&7&0&0&12&0&6&0&12&25&0\\1&0&7&0&2&0&14&19&0&8&0&30&0&0&70\\0&1&0&7&0&13&0&0&16&0&6&0&41&45&0\\0&0&2&0&13&0&18&20&0&28&0&46&0&0&98\\0&7&0&13&0&52&0&0&83&0&40&0&132&202&0\\0&0&14&0&18&0&63&55&0&44&0&130&0&0&294\\5&0&19&0&20&0&55&99&0&67&0&144&0&0&358\\0&12&0&16&0&83&0&0&148&0&69&0&209&348&0\\1&0&8&0&28&0&44&67&0&83&0&126&0&0&302\\0&6&0&6&0&40&0&0&69&0&37&0&95&165&0\\1&0&30&0&46&0&130&144&0&126&0&310&0&0&710\\0&12&0&41&0&132&0&0&209&0&95&0&394&545&0\\0&25&0&45&0&202&0&0&348&0&165&0&545&868&0\\6&0&70&0&98&0&294&358&0&302&0&710&0&0&1712\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&7&7&13&52&63&99&148&83&37&310&394&868&1712&879&899&2136&1808&584\end{bmatrix}$

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$$1/8$$0$$0$$0$$3/8$
$a_1=0$$1/2$$1/4$$0$$0$$0$$0$$1/4$
$a_3=0$$1/2$$1/4$$0$$0$$0$$0$$1/4$
$a_1=a_3=0$$1/2$$1/4$$0$$0$$0$$0$$1/4$