Properties

 Label 1.4.B.1.1a Name $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$ Weight $1$ Degree $4$ Real dimension $6$ Components $1$ Contained in $$\mathrm{USp}(4)$$ Identity component $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$ Component group $$C_1$$

Invariants

 Weight: $1$ Degree: $4$ $\mathbb{R}$-dimension: $6$ Components: $1$ Contained in: $\mathrm{USp}(4)$ Rational: yes

Identity component

 Name: $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $\mathbb{R}$-dimension: $6$ Description: $\left\{\begin{bmatrix}A&0\\0&B\end{bmatrix}:A,B\in\mathrm{SU}(2)\right\}$ Symplectic form: $\begin{bmatrix}J_2&0\\0&J_2\end{bmatrix}, J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ Hodge circle: $u\mapsto\mathrm{diag}(u,\bar u,\bar u,u)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $10$ $0$ $70$ $0$ $588$ $0$ $5544$ $0$ $56628$
$a_2$ $1$ $2$ $5$ $14$ $44$ $152$ $569$ $2270$ $9524$ $41576$ $187348$ $866296$ $4092400$

Moment simplex

 $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $2$ $2$ $5$ $6$ $10$ $14$ $20$ $36$ $70$ $44$ $72$ $138$ $280$ $588$ $152$ $276$ $556$ $1168$ $2520$ $5544$ $569$ $1114$ $2334$ $5044$ $11130$ $24948$ $56628$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&1&0&0&0&1\\0&2&0&0&2&0&0&0&2&0\\1&0&2&1&0&2&0&1&0&2\\0&0&1&3&0&1&0&3&0&1\\0&2&0&0&4&0&2&0&4&0\\1&0&2&1&0&3&0&2&0&3\\0&0&0&0&2&0&4&0&2&0\\0&0&1&3&0&2&0&6&0&2\\0&2&0&0&4&0&2&0&6&0\\1&0&2&1&0&3&0&2&0&4\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&2&3&4&3&4&6&6&4\end{bmatrix}$

Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2$ and $n\in\mathbb{Z}$.

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