# Properties

 Label 1.4.D.4.2a Name $$F_{a,b}$$ Weight $1$ Degree $4$ Real dimension $2$ Components $4$ Contained in $$\mathrm{USp}(4)$$ Identity component $$\mathrm{U}(1)\times\mathrm{U}(1)$$ Component group $$C_2^2$$

## Invariants

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

## Identity component

 Name: $\mathrm{U}(1)\times\mathrm{U}(1)$ $\mathbb{R}$-dimension: $2$ Description: $\left\{\begin{bmatrix}A&0\\0&B\end{bmatrix}:A,B\in \mathrm{U}(1)\subseteq\mathrm{SU}(2)\right\}$ Symplectic form: $\begin{bmatrix}J_2&0\\0&J_2\end{bmatrix}, J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ Hodge circle: $u\mapsto\mathrm{diag}(u,\bar u,\bar u,u)$

## Component group

 Name: $C_2^2$ Order: $4$ Abelian: yes Generators: $\begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}, \begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}$

 Maximal subgroups: $F_a$${}^{\times 2}, F_{ab} Minimal supergroups: ## Moment sequences x \mathrm{E}[x^{0}] \mathrm{E}[x^{1}] \mathrm{E}[x^{2}] \mathrm{E}[x^{3}] \mathrm{E}[x^{4}] \mathrm{E}[x^{5}] \mathrm{E}[x^{6}] \mathrm{E}[x^{7}] \mathrm{E}[x^{8}] \mathrm{E}[x^{9}] \mathrm{E}[x^{10}] \mathrm{E}[x^{11}] \mathrm{E}[x^{12}] a_1 1 0 2 0 12 0 110 0 1260 0 16002 0 213906 a_2 1 2 5 14 49 202 944 4720 24553 130658 705880 3855040 21232468 ## Moment simplex  \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon 2 2 5 6 12 14 22 48 110 49 94 218 520 1260 202 444 1068 2610 6440 16002 944 2224 5466 13560 33852 84924 213906 ## Moment matrix \mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&1&0&0&0&1\\0&2&0&0&2&0&2&0&2&0\\1&0&2&1&0&2&0&3&0&3\\0&0&1&5&0&1&0&7&0&2\\0&2&0&0&6&0&6&0&10&0\\1&0&2&1&0&6&0&7&0&7\\0&2&0&0&6&0&8&0&10&0\\0&0&3&7&0&7&0&19&0&12\\0&2&0&0&10&0&10&0&22&0\\1&0&3&2&0&7&0&12&0&12\end{bmatrix} \ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&2&5&6&6&8&19&22&12\end{bmatrix} ## Event probabilities -$$a_2\in\mathbb{Z}$$a_2=-2$$a_2=-1$$a_2=0$$a_2=1$$a_2=2 -$$1$$3/4$$0$$0$$0$$0$$3/4$
$a_1=0$$1/4$$1/4$$0$$0$$0$$0$$1/4$