Properties

Label 1.6.N.12.5b
  
Name \(A(6,2)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $12$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_2\times C_6\)

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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:$C_2\times C_6$
Order:$12$
Abelian:yes
Generators:$\begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $6$ $0$ $90$ $0$ $1860$ $0$ $47250$ $0$ $1369116$ $0$ $42813540$
$a_2$ $1$ $3$ $21$ $183$ $1881$ $21723$ $271935$ $3588273$ $48949713$ $681841731$ $9625504491$ $137096373813$ $1964839795527$
$a_3$ $1$ $0$ $32$ $0$ $5160$ $0$ $1388060$ $0$ $455514808$ $0$ $160776717912$ $0$ $58526023915524$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $3$ $6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $21$ $12$ $42$ $90$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $32$ $183$ $108$ $390$ $228$ $846$ $1860$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $288$ $1881$ $1080$ $630$ $4158$ $2382$ $9288$ $5280$ $20880$ $47250$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $3096$ $21723$ $1764$ $12216$ $6906$ $49254$ $27588$ $15540$ $112356$ $62676$ $257520$ $143052$ $592662$ $1369116$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $5160$ $36696$ $271935$ $20556$ $150564$ $83658$ $627030$ $46632$ $346122$ $191628$ $1451250$ $798792$ $440916$ $3369420$
$$ $1849848$ $7844382$ $4296600$ $18306540$ $42813540$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&3&0&4&6&0&4&0&9&0&0&16\\0&6&0&6&0&24&0&0&30&0&18&0&42&66&0\\2&0&16&0&18&0&44&48&0&32&0&90&0&0&188\\0&6&0&14&0&40&0&0&54&0&26&0&96&128&0\\3&0&18&0&27&0&54&66&0&54&0&123&0&0&264\\0&24&0&40&0&152&0&0&228&0&112&0&360&544&0\\4&0&44&0&54&0&160&174&0&136&0&348&0&0&808\\6&0&48&0&66&0&174&222&0&186&0&414&0&0&996\\0&30&0&54&0&228&0&0&390&0&180&0&606&960&0\\4&0&32&0&54&0&136&186&0&174&0&348&0&0&872\\0&18&0&26&0&112&0&0&180&0&92&0&276&440&0\\9&0&90&0&123&0&348&414&0&348&0&825&0&0&1992\\0&42&0&96&0&360&0&0&606&0&276&0&1032&1548&0\\0&66&0&128&0&544&0&0&960&0&440&0&1548&2450&0\\16&0&188&0&264&0&808&996&0&872&0&1992&0&0&5028\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&6&16&14&27&152&160&222&390&174&92&825&1032&2450&5028&2634&2649&6620&5880&1798\end{bmatrix}$

Event probabilities

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