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Properties

Label 1.6.N.36.10a
  
Name \(J(D(3,1))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $36$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(S_3^2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$36$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)_3$
$\mathbb{R}$-dimension:$1$
Description:$\left\{\begin{bmatrix}\alpha I_3&0,\\ 0&\bar \alpha I_3\end{bmatrix}: \alpha\bar\alpha=1,\ \alpha\in\mathbb{C}\right\}$ Symplectic form:$\begin{bmatrix} 0 & I_3\\ -I_3 & 0\end{bmatrix}$
Hodge circle:$u\mapsto \mathrm{diag}(u,u, u, \bar u, \bar u, \bar u)$

Component group

Name:$S_3^2$
Order:$36$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{3}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{2} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0 & 0& 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(C(3,1))$${}^{\times 2}$, $J(B(3,1))$${}^{\times 2}$, $D(3,1)$
Minimal supergroups:$J(D(6,2))$, $J(D(3,3))$, $J(E(36))$${}^{\times 2}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $15$ $0$ $410$ $0$ $12775$ $0$ $413406$ $0$ $13640550$
$a_2$ $1$ $1$ $5$ $40$ $457$ $5946$ $80934$ $1123305$ $15768729$ $223135474$ $3176981140$ $45458037615$ $653100662218$
$a_3$ $1$ $0$ $6$ $0$ $1322$ $0$ $429450$ $0$ $148945538$ $0$ $53333020896$ $0$ $19484986431482$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $5$ $1$ $6$ $15$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $6$ $40$ $19$ $77$ $40$ $175$ $410$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $57$ $457$ $240$ $135$ $1013$ $555$ $2350$ $1275$ $5465$ $12775$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $758$ $5946$ $408$ $3214$ $1751$ $13731$ $7478$ $4090$ $32076$ $17461$ $75090$ $40796$ $176043$ $413406$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1322$ $10235$ $80934$ $5582$ $43917$ $23893$ $189300$ $12982$ $102831$ $55880$ $444327$ $241173$ $131026$ $1044291$
$$ $566454$ $2457105$ $1331778$ $5786760$ $13640550$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&0&3&0&1&0&1&0&0&4\\0&1&0&0&0&4&0&0&8&0&4&0&8&19&0\\0&0&4&0&1&0&9&13&0&6&0&21&0&0&56\\0&0&0&5&0&8&0&0&12&0&4&0&32&35&0\\0&0&1&0&8&0&11&15&0&22&0&33&0&0&80\\0&4&0&8&0&36&0&0&64&0&28&0&104&164&0\\0&0&9&0&11&0&48&42&0&35&0&102&0&0&252\\3&0&13&0&15&0&42&75&0&59&0&119&0&0&316\\0&8&0&12&0&64&0&0&121&0&56&0&180&307&0\\1&0&6&0&22&0&35&59&0&71&0&110&0&0&280\\0&4&0&4&0&28&0&0&56&0&28&0&76&140&0\\1&0&21&0&33&0&102&119&0&110&0&253&0&0&640\\0&8&0&32&0&104&0&0&180&0&76&0&344&488&0\\0&19&0&35&0&164&0&0&307&0&140&0&488&799&0\\4&0&56&0&80&0&252&316&0&280&0&640&0&0&1644\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&4&5&8&36&48&75&121&71&28&253&344&799&1644&891&901&2236&1974&655\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$13/18$$0$$5/9$$0$$0$$1/6$
$a_1=0$$13/18$$13/18$$0$$5/9$$0$$0$$1/6$
$a_3=0$$1/2$$1/2$$0$$1/3$$0$$0$$1/6$
$a_1=a_3=0$$1/2$$1/2$$0$$1/3$$0$$0$$1/6$