Properties

Label 1.6.N.12.4d
  
Name \(J(A(3,2))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $12$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(D_6\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$12$
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:$D_6$
Order:$12$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0& 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(A(1,2))$, $A(3,2)$, $J(A(3,1))$${}^{\times 2}$
Minimal supergroups:$J(A(3,4))$${}^{\times 2}$, $J(B(3,2;2))$, $J(A(6,2))$${}^{\times 6}$, $J(A(3,6))$, $J_s(B(3,2))$, $J(B(3,2))$, $J_s(B(T,1))$, $J_s(B(3,2;2))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $57$ $0$ $1430$ $0$ $41265$ $0$ $1283058$ $0$ $41551818$
$a_2$ $1$ $3$ $17$ $142$ $1545$ $19148$ $252564$ $3442869$ $47854569$ $673553236$ $9562469142$ $136614883199$ $1961147974624$
$a_3$ $1$ $0$ $20$ $0$ $4340$ $0$ $1321290$ $0$ $449725276$ $0$ $160258222860$ $0$ $58478646649346$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $17$ $7$ $26$ $57$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $20$ $142$ $73$ $283$ $160$ $633$ $1430$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $211$ $1545$ $840$ $473$ $3411$ $1901$ $7794$ $4325$ $17895$ $41265$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $2546$ $19148$ $1416$ $10498$ $5809$ $43877$ $24154$ $13334$ $101604$ $55803$ $236014$ $129332$ $549661$ $1283058$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $4340$ $32701$ $252564$ $18006$ $138019$ $75667$ $587604$ $41554$ $321033$ $175656$ $1372401$ $748619$ $408926$ $3211725$
$$ $1749482$ $7528983$ $4095966$ $17675784$ $41551818$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&2&7&0&1&0&5&0&0&14\\0&3&0&4&0&16&0&0&24&0&12&0&36&57&0\\2&0&12&0&9&0&31&45&0&26&0&71&0&0&174\\0&4&0&9&0&30&0&0&48&0&22&0&80&119&0\\0&0&9&0&22&0&47&45&0&52&0&109&0&0&248\\0&16&0&30&0&122&0&0&204&0&94&0&328&510&0\\2&0&31&0&47&0&136&152&0&137&0&318&0&0&774\\7&0&45&0&45&0&152&217&0&159&0&377&0&0&964\\0&24&0&48&0&204&0&0&363&0&164&0&584&925&0\\1&0&26&0&52&0&137&159&0&163&0&342&0&0&852\\0&12&0&22&0&94&0&0&164&0&78&0&264&418&0\\5&0&71&0&109&0&318&377&0&342&0&781&0&0&1944\\0&36&0&80&0&328&0&0&584&0&264&0&976&1516&0\\0&57&0&119&0&510&0&0&925&0&418&0&1516&2403&0\\14&0&174&0&248&0&774&964&0&852&0&1944&0&0&4962\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&12&9&22&122&136&217&363&163&78&781&976&2403&4962&2581&2675&6558&5850&1731\end{bmatrix}$

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