Properties

Label 1.6.B.2.1a
  
Name \(N(\mathrm{U}(3))\)
Weight $1$
Degree $6$
Real dimension $9$
Components $2$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(3)\)
Component group \(C_2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$9$
Components:$2$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(3)$
$\mathbb{R}$-dimension:$9$
Description:$\left\{\begin{bmatrix}A&0\\0&\overline{A}\end{bmatrix}: A\in\mathrm{U}(3)\right\}$ 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:$C_2$
Order:$2$
Abelian:yes
Generators:$\begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 &0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0\\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$\mathrm{U}(3)$
Minimal supergroups:

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $6$ $0$ $60$ $0$ $805$ $0$ $12978$ $0$ $237006$
$a_2$ $1$ $1$ $3$ $11$ $58$ $382$ $2922$ $24642$ $222728$ $2122232$ $21092807$ $217061737$ $2300105315$
$a_3$ $1$ $0$ $3$ $0$ $116$ $0$ $10630$ $0$ $1422428$ $0$ $236405736$ $0$ $45382171296$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $3$ $1$ $3$ $6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $3$ $11$ $6$ $16$ $10$ $30$ $60$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $12$ $58$ $34$ $23$ $101$ $65$ $198$ $125$ $395$ $805$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $77$ $382$ $48$ $233$ $147$ $735$ $460$ $292$ $1482$ $925$ $3026$ $1876$ $6237$ $12978$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $116$ $554$ $2922$ $348$ $1785$ $1107$ $5908$ $687$ $3631$ $2241$ $12189$ $7464$ $4597$ $25356$
$$ $15477$ $53109$ $32298$ $111888$ $237006$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&0&1&0&0&0&0&0&0&0\\0&1&0&0&0&1&0&0&1&0&1&0&0&1&0\\0&0&2&0&0&0&1&1&0&0&0&2&0&0&2\\0&0&0&2&0&1&0&0&0&0&0&0&3&1&0\\0&0&0&0&3&0&1&1&0&2&0&2&0&0&2\\0&1&0&1&0&4&0&0&3&0&2&0&4&5&0\\0&0&1&0&1&0&5&1&0&0&0&4&0&0&6\\1&0&1&0&1&0&1&6&0&2&0&3&0&0&6\\0&1&0&0&0&3&0&0&6&0&2&0&3&6&0\\0&0&0&0&2&0&0&2&0&6&0&2&0&0&4\\0&1&0&0&0&2&0&0&2&0&3&0&1&4&0\\0&0&2&0&2&0&4&3&0&2&0&9&0&0&10\\0&0&0&3&0&4&0&0&3&0&1&0&11&7&0\\0&1&0&1&0&5&0&0&6&0&4&0&7&13&0\\0&0&2&0&2&0&6&6&0&4&0&10&0&0&20\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&2&3&4&5&6&6&6&3&9&11&13&20&12&11&20&16&10\end{bmatrix}$

Event probabilities

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