Invariants
| Weight: | $1$ |
| Degree: | $2$ |
| $\mathbb{R}$-dimension: | $1$ |
| Components: | $4$ |
| Contained in: | $\mathrm{U}(2)$ |
| Rational: | yes |
Identity component
| Name: | $\mathrm{U}(1)$ | ||
| $\mathbb{R}$-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: | $D_{2}$ |
| Order: | $4$ |
| Abelian: | yes |
| 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 sequences
| $x$ | $\mathrm{E}[x^{0}]$ | $\mathrm{E}[x^{1}]$ | $\mathrm{E}[x^{2}]$ | $\mathrm{E}[x^{3}]$ | $\mathrm{E}[x^{4}]$ | $\mathrm{E}[x^{5}]$ | $\mathrm{E}[x^{6}]$ | $\mathrm{E}[x^{7}]$ | $\mathrm{E}[x^{8}]$ | $\mathrm{E}[x^{9}]$ | $\mathrm{E}[x^{10}]$ | $\mathrm{E}[x^{11}]$ | $\mathrm{E}[x^{12}]$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $a_1$ | $1$ | $0$ | $1$ | $0$ | $3$ | $0$ | $10$ | $0$ | $35$ | $0$ | $126$ | $0$ | $462$ |