Name: | $\mathrm{SU}(2)\times\mathrm{SU}(2)_2$ |
$\mathbb{R}$-dimension: | $6$ |
Description: | $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&\overline{B}\end{bmatrix}: A,B\in\mathrm{SU}(2)\right\}$ |
Symplectic form: | $\begin{bmatrix}J_2&0&0\\0&0&I_2\\0&-I_2&0\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ |
Hodge circle: | $\mathrm{diag}(u,\bar u, u,\bar u,\bar u,u)$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$5$ |
$0$ |
$58$ |
$0$ |
$925$ |
$0$ |
$17598$ |
$0$ |
$374850$ |
$0$ |
$8638740$ |
$a_2$ |
$1$ |
$4$ |
$21$ |
$137$ |
$1059$ |
$9250$ |
$88075$ |
$892540$ |
$9478251$ |
$104393828$ |
$1184015783$ |
$13756977657$ |
$163109827813$ |
$a_3$ |
$1$ |
$0$ |
$26$ |
$0$ |
$2444$ |
$0$ |
$366740$ |
$0$ |
$70009940$ |
$0$ |
$15468784248$ |
$0$ |
$3769925528688$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$4$ |
$5$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$21$ |
$11$ |
$32$ |
$58$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$26$ |
$137$ |
$77$ |
$241$ |
$142$ |
$464$ |
$925$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$191$ |
$1059$ |
$612$ |
$360$ |
$2037$ |
$1198$ |
$4102$ |
$2415$ |
$8436$ |
$17598$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1586$ |
$9250$ |
$930$ |
$5382$ |
$3154$ |
$18775$ |
$10976$ |
$6434$ |
$39030$ |
$22793$ |
$82221$ |
$47922$ |
$174888$ |
$374850$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2444$ |
$14433$ |
$88075$ |
$8430$ |
$51127$ |
$29780$ |
$184830$ |
$17385$ |
$107300$ |
$62411$ |
$393102$ |
$227796$ |
$132232$ |
$843417$ |
$$ |
$487746$ |
$1821942$ |
$1051470$ |
$3957840$ |
$8638740$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&3&0&1&0&3&6&0&1&0&3&0&0&8\\0&5&0&6&0&16&0&0&18&0&5&0&17&33&0\\3&0&14&0&10&0&25&33&0&14&0&34&0&0&72\\0&6&0&9&0&24&0&0&30&0&9&0&32&54&0\\1&0&10&0&15&0&29&28&0&21&0&50&0&0&88\\0&16&0&24&0&76&0&0&96&0&40&0&120&184&0\\3&0&25&0&29&0&70&75&0&51&0&117&0&0&232\\6&0&33&0&28&0&75&96&0&52&0&124&0&0&264\\0&18&0&30&0&96&0&0&133&0&52&0&169&258&0\\1&0&14&0&21&0&51&52&0&44&0&94&0&0&192\\0&5&0&9&0&40&0&0&52&0&35&0&85&110&0\\3&0&34&0&50&0&117&124&0&94&0&235&0&0&448\\0&17&0&32&0&120&0&0&169&0&85&0&251&350&0\\0&33&0&54&0&184&0&0&258&0&110&0&350&535&0\\8&0&72&0&88&0&232&264&0&192&0&448&0&0&940\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&5&14&9&15&76&70&96&133&44&35&235&251&535&940&399&449&887&642&155\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.