# Properties

 Label 1.4.E.4.2a Name $$J(E_2)$$ Weight $1$ Degree $4$ Real dimension $3$ Components $4$ Contained in $$\mathrm{USp}(4)$$ Identity component $$\mathrm{SU}(2)_2$$ Component group $$C_2^2$$

## Invariants

 Weight: $1$ Degree: $4$ $\mathbb{R}$-dimension: $3$ Components: $4$ Contained in: $\mathrm{USp}(4)$ Rational: yes

## Identity component

 Name: $\mathrm{SU}(2)_2$ $\mathbb{R}$-dimension: $3$ Description: $\left\{\begin{bmatrix}A&0\\0&\bar{A}\end{bmatrix}: A\in \mathrm{SU}(2)\right\}$ Symplectic form: $\begin{bmatrix}0&I_2\\-I_2&0\end{bmatrix}$ Hodge circle: $u\mapsto\mathrm{diag}(u,\bar u,\bar u,u)$

## Component group

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

 Maximal subgroups: $E_2$, $J(E_1)$${}^{\times 2} Minimal supergroups: J(E_4), J(E_6) ## Moment sequences x \mathrm{E}[x^{0}] \mathrm{E}[x^{1}] \mathrm{E}[x^{2}] \mathrm{E}[x^{3}] \mathrm{E}[x^{4}] \mathrm{E}[x^{5}] \mathrm{E}[x^{6}] \mathrm{E}[x^{7}] \mathrm{E}[x^{8}] \mathrm{E}[x^{9}] \mathrm{E}[x^{10}] \mathrm{E}[x^{11}] \mathrm{E}[x^{12}] a_1 1 0 1 0 8 0 80 0 896 0 10752 0 135168 a_2 1 1 4 10 42 166 768 3620 17902 90310 465096 2429164 12843988 ## Moment simplex  \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon 1 1 4 4 8 10 17 36 80 42 76 168 384 896 166 354 808 1888 4480 10752 768 1704 3984 9472 22784 55296 135168 ## Moment matrix \mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&2&0&0&0&0\\0&1&0&0&2&0&1&0&3&0\\0&0&3&0&0&0&0&4&0&6\\0&0&0&4&0&3&0&4&0&1\\0&2&0&0&5&0&4&0&8&0\\2&0&0&3&0&8&0&2&0&0\\0&1&0&0&4&0&5&0&7&0\\0&0&4&4&0&2&0&15&0&12\\0&3&0&0&8&0&7&0&14&0\\0&0&6&1&0&0&0&12&0&16\end{bmatrix} \ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&3&4&5&8&5&15&14&16\end{bmatrix} ## Event probabilities -$$a_2\in\mathbb{Z}$$a_2=-2$$a_2=-1$$a_2=0$$a_2=1$$a_2=2 -$$1$$0$$0$$0$$0$$0$$0$
$a_1=0$$3/4$$0$$0$$0$$0$$0$$0$