Properties

Label 1.6.N.6.1b
  
Name \(J(A(3,1))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $6$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(S_3\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$6$
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:$S_3$
Order:$6$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{3}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{2} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0& 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(A(1,1))$, $A(3,1)$
Minimal supergroups:$J(C(3,1))$, $J(B(3,1))$, $J(A(3,2))$${}^{\times 2}$, $J_s(C(2,2))$, $J(A(3,3))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $81$ $0$ $2430$ $0$ $76545$ $0$ $2480058$ $0$ $81841914$
$a_2$ $1$ $3$ $21$ $216$ $2673$ $35478$ $485028$ $6738147$ $94607433$ $1338798294$ $19061843886$ $272748098637$ $3918603596904$
$a_3$ $1$ $0$ $28$ $0$ $7860$ $0$ $2575810$ $0$ $893661020$ $0$ $319997950668$ $0$ $116909916032514$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $21$ $9$ $36$ $81$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $28$ $216$ $111$ $459$ $252$ $1053$ $2430$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $345$ $2673$ $1440$ $789$ $6075$ $3321$ $14094$ $7695$ $32805$ $76545$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $4542$ $35478$ $2484$ $19278$ $10521$ $82377$ $44874$ $24462$ $192456$ $104733$ $450522$ $244944$ $1056321$ $2480058$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $7860$ $61407$ $485028$ $33462$ $263493$ $143343$ $1135782$ $78030$ $616977$ $335340$ $2665953$ $1447065$ $785862$ $6265755$
$$ $3398598$ $14742567$ $7991298$ $34720812$ $81841914$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&4&9&0&3&0&9&0&0&26\\0&3&0&6&0&24&0&0&42&0&18&0&66&105&0\\2&0&16&0&15&0&53&75&0&54&0&129&0&0&334\\0&6&0&13&0&50&0&0&90&0&40&0&144&229&0\\0&0&15&0&30&0&81&87&0&90&0&195&0&0&480\\0&24&0&50&0&214&0&0&384&0&170&0&624&986&0\\4&0&53&0&81&0&244&294&0&267&0&600&0&0&1514\\9&0&75&0&87&0&294&393&0&315&0&735&0&0&1896\\0&42&0&90&0&384&0&0&699&0&312&0&1146&1815&0\\3&0&54&0&90&0&267&315&0&299&0&666&0&0&1684\\0&18&0&40&0&170&0&0&312&0&142&0&516&814&0\\9&0&129&0&195&0&600&735&0&666&0&1509&0&0&3840\\0&66&0&144&0&624&0&0&1146&0&516&0&1896&3000&0\\0&105&0&229&0&986&0&0&1815&0&814&0&3000&4759&0\\26&0&334&0&480&0&1514&1896&0&1684&0&3840&0&0&9858\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&16&13&30&214&244&393&699&299&142&1509&1896&4759&9858&5109&5295&13054&11634&3395\end{bmatrix}$

Event probabilities

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