Properties

Label 8.2.1.d2
  
Name \(\mathrm{U}(1)[D_{2}]\)
Weight 8
Degree 2
Real dimension 1
Components 4
Contained in \(\mathrm{U}(2)\)
Identity Component \(\mathrm{U}(1)\)
Component group \(D_{2}\)

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Invariants

Weight:$8$
Degree:$2$
$\mathbb{R}$-dimension:$1$
Components:$4$
Contained in:$\mathrm{U}(2)$
Rational:$\mathrm{True}$

Identity Component

Name:$\mathrm{U}(1)$
Index:$4$
$\mathbb{R}$-dimension:$1$
Description:$\left\{\begin{bmatrix}\alpha&0\\0&\bar\alpha\end{bmatrix}:\alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$

Component Group

Name:$D_{2}$
Order:$4$
Abelian:$\mathrm{True}$
Generators:$\left\{\begin{bmatrix} 0 & 1\\ -1 & 0\end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & \zeta_{2}\end{bmatrix}\right\}$

Subgroups and Supergroups

Maximal Subgroups:$N(\mathrm{U}(1))$
Minimal Supergroups:$\mathrm{U}(1)[D_{4}]$, $\mathrm{U}(1)[D_{6}]$, $\mathrm{U}(1)[D_{10}]$, $\ldots$

Moment Statistics

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $3$ $0$ $10$ $0$ $35$ $0$ $126$ $0$ $462$