Properties

Label 1.6.B.1.1a
  
Name \(\mathrm{U}(3)\)
Weight $1$
Degree $6$
Real dimension $9$
Components $1$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(3)\)
Component group \(C_1\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$9$
Components:$1$
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:$u\mapsto\mathrm{diag}(u,u,u,\bar u,\bar u,\bar u)$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $12$ $0$ $120$ $0$ $1610$ $0$ $25956$ $0$ $474012$
$a_2$ $1$ $1$ $4$ $18$ $107$ $743$ $5793$ $49157$ $445133$ $4243629$ $42183426$ $434117676$ $4600195119$
$a_3$ $1$ $0$ $6$ $0$ $232$ $0$ $21260$ $0$ $2844856$ $0$ $472811472$ $0$ $90764342592$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $4$ $2$ $6$ $12$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $6$ $18$ $12$ $32$ $20$ $60$ $120$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $24$ $107$ $68$ $46$ $202$ $130$ $396$ $250$ $790$ $1610$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $154$ $743$ $96$ $466$ $294$ $1470$ $920$ $584$ $2964$ $1850$ $6052$ $3752$ $12474$ $25956$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $232$ $1108$ $5793$ $696$ $3570$ $2214$ $11816$ $1374$ $7262$ $4482$ $24378$ $14928$ $9194$ $50712$
$$ $30954$ $106218$ $64596$ $223776$ $474012$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&1&0&0&1&0&1&0&0&0&0&0\\0&2&0&0&0&2&0&0&2&0&2&0&0&2&0\\0&0&3&0&1&0&3&1&0&0&0&4&0&0&4\\0&0&0&4&0&2&0&0&0&0&0&0&6&2&0\\1&0&1&0&4&0&1&4&0&3&0&4&0&0&4\\0&2&0&2&0&8&0&0&6&0&4&0&8&10&0\\0&0&3&0&1&0&9&3&0&0&0&8&0&0&12\\1&0&1&0&4&0&3&9&0&6&0&7&0&0&12\\0&2&0&0&0&6&0&0&12&0&4&0&6&12&0\\1&0&0&0&3&0&0&6&0&10&0&3&0&0&8\\0&2&0&0&0&4&0&0&4&0&6&0&2&8&0\\0&0&4&0&4&0&8&7&0&3&0&17&0&0&20\\0&0&0&6&0&8&0&0&6&0&2&0&22&14&0\\0&2&0&2&0&10&0&0&12&0&8&0&14&26&0\\0&0&4&0&4&0&12&12&0&8&0&20&0&0&40\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&3&4&4&8&9&9&12&10&6&17&22&26&40&24&19&40&30&20\end{bmatrix}$

Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.