Properties

Label 1.6.L.12.4c
  
Name \(L(D_{6,1},D_3)\)
Weight $1$
Degree $6$
Real dimension $6$
Components $12$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{U}(1)_2\)
Component group \(D_6\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$6$
Components:$12$
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:$D_6$
Order:$12$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{1} \\0 & 0 & 0 & 0 & \zeta_{12}^{5} & 0 \\0 & 0 & 0 & \zeta_{12}^{5} & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$L_1(D_3)$, $L(D_{2,1},C_2)$, $L(D_{3,2},C_3)$, $L(C_{6,1},C_3)$
Minimal supergroups:$L_2(D_{6,1})$, $L(J(D_6),D_6)$${}^{\times 2}$, $L(J(D_6),J(D_3))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $24$ $0$ $480$ $0$ $12040$ $0$ $336672$ $0$ $9991212$
$a_2$ $1$ $2$ $9$ $57$ $511$ $5577$ $67096$ $849081$ $11072067$ $147290769$ $1987765354$ $27120975741$ $373258539110$
$a_3$ $1$ $0$ $9$ $0$ $1311$ $0$ $330030$ $0$ $98092799$ $0$ $31411678914$ $0$ $10494859447842$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $9$ $3$ $11$ $24$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $9$ $57$ $28$ $100$ $57$ $216$ $480$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $74$ $511$ $276$ $161$ $1061$ $606$ $2365$ $1340$ $5315$ $12040$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $790$ $5577$ $447$ $3096$ $1750$ $12380$ $6958$ $3930$ $28092$ $15750$ $64056$ $35805$ $146608$ $336672$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1311$ $9200$ $67096$ $5172$ $37276$ $20836$ $152876$ $11664$ $85074$ $47436$ $350946$ $194916$ $108487$ $807992$
$$ $447986$ $1864758$ $1032192$ $4312476$ $9991212$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&0&0&1&0&0&4\\0&2&0&1&0&6&0&0&8&0&5&0&9&18&0\\1&0&6&0&2&0&10&16&0&5&0&21&0&0&48\\0&1&0&5&0&10&0&0&13&0&5&0&27&32&0\\0&0&2&0&11&0&14&12&0&19&0&35&0&0&66\\0&6&0&10&0&39&0&0&58&0&29&0&91&139&0\\0&0&10&0&14&0&45&39&0&34&0&90&0&0&200\\4&0&16&0&12&0&39&71&0&40&0&98&0&0&242\\0&8&0&13&0&58&0&0&100&0&45&0&147&234&0\\0&0&5&0&19&0&34&40&0&52&0&87&0&0&204\\0&5&0&5&0&29&0&0&45&0&27&0&66&112&0\\1&0&21&0&35&0&90&98&0&87&0&213&0&0&476\\0&9&0&27&0&91&0&0&147&0&66&0&260&369&0\\0&18&0&32&0&139&0&0&234&0&112&0&369&583&0\\4&0&48&0&66&0&200&242&0&204&0&476&0&0&1150\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&5&11&39&45&71&100&52&27&213&260&583&1150&580&607&1401&1196&355\end{bmatrix}$

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$$0$$1/6$$0$$0$$1/3$
$a_1=0$$1/2$$1/2$$0$$1/6$$0$$0$$1/3$
$a_3=0$$1/2$$1/2$$0$$1/6$$0$$0$$1/3$
$a_1=a_3=0$$1/2$$1/2$$0$$1/6$$0$$0$$1/3$