Properties

Label 1.6.N.6.2a
  
Name \(J_s(A(3,1))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $6$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_6\)

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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:$C_6$
Order:$6$
Abelian:yes
Generators:$\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{3}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{2} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(A(1,1))$, $A(3,1)$
Minimal supergroups:$J_s(A(3,3))$, $J(C(3,1))$, $J(C(1,7))$, $J(B(3,1))$, $J_s(A(3,2))$${}^{\times 2}$, $J(C(2,2))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $81$ $0$ $2430$ $0$ $76545$ $0$ $2480058$ $0$ $81841914$
$a_2$ $1$ $2$ $18$ $207$ $2646$ $35397$ $484785$ $6737418$ $94605246$ $1338791733$ $19061824203$ $272748039588$ $3918603419757$
$a_3$ $1$ $0$ $28$ $0$ $7860$ $0$ $2575810$ $0$ $893661020$ $0$ $319997950668$ $0$ $116909916032514$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $18$ $9$ $36$ $81$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $28$ $207$ $111$ $459$ $252$ $1053$ $2430$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $345$ $2646$ $1440$ $789$ $6075$ $3321$ $14094$ $7695$ $32805$ $76545$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $4542$ $35397$ $2484$ $19278$ $10521$ $82377$ $44874$ $24462$ $192456$ $104733$ $450522$ $244944$ $1056321$ $2480058$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $7860$ $61407$ $484785$ $33462$ $263493$ $143343$ $1135782$ $78030$ $616977$ $335340$ $2665953$ $1447065$ $785862$ $6265755$
$$ $3398598$ $14742567$ $7991298$ $34720812$ $81841914$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&5&7&0&4&0&10&0&0&26\\0&3&0&6&0&24&0&0&42&0&18&0&66&105&0\\1&0&15&0&17&0&54&71&0&57&0&131&0&0&334\\0&6&0&13&0&50&0&0&90&0&40&0&144&229&0\\1&0&17&0&27&0&79&93&0&86&0&192&0&0&480\\0&24&0&50&0&214&0&0&384&0&170&0&624&986&0\\5&0&54&0&79&0&243&298&0&264&0&598&0&0&1514\\7&0&71&0&93&0&298&381&0&323&0&741&0&0&1896\\0&42&0&90&0&384&0&0&699&0&312&0&1146&1815&0\\4&0&57&0&86&0&264&323&0&294&0&662&0&0&1684\\0&18&0&40&0&170&0&0&312&0&142&0&516&814&0\\10&0&131&0&192&0&598&741&0&662&0&1506&0&0&3840\\0&66&0&144&0&624&0&0&1146&0&516&0&1896&3000&0\\0&105&0&229&0&986&0&0&1815&0&814&0&3000&4759&0\\26&0&334&0&480&0&1514&1896&0&1684&0&3840&0&0&9858\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&15&13&27&214&243&381&699&294&142&1506&1896&4759&9858&5109&5268&13054&11622&3395\end{bmatrix}$

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$$0$$2/3$$0$$0$$1/6$
$a_1=0$$5/6$$5/6$$0$$2/3$$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$