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Properties

Label 1.6.N.3.1a
  
Name \(A(3,1)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $3$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_3\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$3$
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:$C_3$
Order:$3$
Abelian:yes
Generators:$\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{3}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{2} \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$A(1,1)$
Minimal supergroups:$B(3,1)$, $C(1,7)$, $C(2,2)$, $A(3,2)$, $C(3,1)$${}^{\times 4}$, $J(A(3,1))$, $A(3,3)$, $J_s(A(3,1))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $6$ $0$ $162$ $0$ $4860$ $0$ $153090$ $0$ $4960116$ $0$ $163683828$
$a_2$ $1$ $3$ $33$ $405$ $5265$ $70713$ $969327$ $13474107$ $189208305$ $2677576905$ $38123628723$ $545496020127$ $7837206662367$
$a_3$ $1$ $0$ $56$ $0$ $15720$ $0$ $5151620$ $0$ $1787322040$ $0$ $639995901336$ $0$ $233819832065028$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $33$ $18$ $72$ $162$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $56$ $405$ $222$ $918$ $504$ $2106$ $4860$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $690$ $5265$ $2880$ $1578$ $12150$ $6642$ $28188$ $15390$ $65610$ $153090$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $9084$ $70713$ $4968$ $38556$ $21042$ $164754$ $89748$ $48924$ $384912$ $209466$ $901044$ $489888$ $2112642$ $4960116$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $15720$ $122814$ $969327$ $66924$ $526986$ $286686$ $2271564$ $156060$ $1233954$ $670680$ $5331906$ $2894130$ $1571724$ $12531510$
$$ $6797196$ $29485134$ $15982596$ $69441624$ $163683828$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&3&0&10&12&0&10&0&21&0&0&52\\0&6&0&12&0&48&0&0&84&0&36&0&132&210&0\\2&0&28&0&36&0&110&138&0&116&0&264&0&0&668\\0&12&0&26&0&100&0&0&180&0&80&0&288&458&0\\3&0&36&0&51&0&156&192&0&168&0&381&0&0&960\\0&48&0&100&0&428&0&0&768&0&340&0&1248&1972&0\\10&0&110&0&156&0&484&600&0&526&0&1194&0&0&3028\\12&0&138&0&192&0&600&750&0&654&0&1488&0&0&3792\\0&84&0&180&0&768&0&0&1398&0&624&0&2292&3630&0\\10&0&116&0&168&0&526&654&0&582&0&1320&0&0&3368\\0&36&0&80&0&340&0&0&624&0&284&0&1032&1628&0\\21&0&264&0&381&0&1194&1488&0&1320&0&3009&0&0&7680\\0&132&0&288&0&1248&0&0&2292&0&1032&0&3792&6000&0\\0&210&0&458&0&1972&0&0&3630&0&1628&0&6000&9518&0\\52&0&668&0&960&0&3028&3792&0&3368&0&7680&0&0&19716\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&6&28&26&51&428&484&750&1398&582&284&3009&3792&9518&19716&10218&10509&26108&23232&6790\end{bmatrix}$

Event probabilities

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