Properties

Label 6.2.B.d9
  
Name \(\mathrm{U}(1)[D_{9}]\)
Weight $6$
Degree $2$
Real dimension $1$
Components $18$
Contained in \(\mathrm{U}(2)\)
Identity component \(\mathrm{U}(1)\)
Component group \(D_{9}\)

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Invariants

Weight:$6$
Degree:$2$
$\mathbb{R}$-dimension:$1$
Components:$18$
Contained in:$\mathrm{U}(2)$
Rational:no

Identity component

Name:$\mathrm{U}(1)$
Index:$18$
$\mathbb{R}$-dimension:$1$
Description:$\left\{\begin{bmatrix}\alpha&0\\0&\bar\alpha\end{bmatrix}:\alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ Symplectic form:$\begin{bmatrix}0&1\\-1&0\end{bmatrix}$
Hodge circle:$u\mapsto \mathrm{diag}(u,u^{-1})$

Component group

Name:$D_{9}$
Order:$18$
Abelian:no
Generators:$\left\{\begin{bmatrix} 0 & 1\\ -1 & 0\end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & \zeta_{9}\end{bmatrix}\right\}$

Subgroups and supergroups

Maximal subgroups:$\mathrm{U}(1)[D_{3}]$
Minimal supergroups:$\mathrm{U}(1)[D_{18}]$, $\mathrm{U}(1)[D_{27}]$, $\mathrm{U}(1)[D_{45}]$, $\ldots$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$