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: | $\mathrm{diag}(u,\bar u, u,\bar u,\bar u,u)$ |
Name: | $C_2$ |
Order: | $2$ |
Abelian: | yes |
Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$30$ |
$0$ |
$465$ |
$0$ |
$8806$ |
$0$ |
$187446$ |
$0$ |
$4319436$ |
$a_2$ |
$1$ |
$3$ |
$13$ |
$75$ |
$547$ |
$4673$ |
$44171$ |
$446645$ |
$4740187$ |
$52199937$ |
$592016543$ |
$6878513689$ |
$81554985589$ |
$a_3$ |
$1$ |
$0$ |
$14$ |
$0$ |
$1228$ |
$0$ |
$183420$ |
$0$ |
$35005460$ |
$0$ |
$7734397416$ |
$0$ |
$1884962825328$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$3$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$13$ |
$6$ |
$17$ |
$30$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$14$ |
$75$ |
$40$ |
$123$ |
$72$ |
$234$ |
$465$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$98$ |
$547$ |
$310$ |
$182$ |
$1025$ |
$602$ |
$2056$ |
$1210$ |
$4223$ |
$8806$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$800$ |
$4673$ |
$468$ |
$2702$ |
$1582$ |
$9405$ |
$5496$ |
$3222$ |
$19528$ |
$11404$ |
$41123$ |
$23968$ |
$87458$ |
$187446$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1228$ |
$7236$ |
$44171$ |
$4224$ |
$25594$ |
$14904$ |
$92463$ |
$8700$ |
$53672$ |
$31218$ |
$196586$ |
$113918$ |
$66130$ |
$421741$ |
$$ |
$243894$ |
$911006$ |
$525756$ |
$1978962$ |
$4319436$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&4&0&0&0&1&0&0&4\\0&3&0&3&0&8&0&0&9&0&2&0&8&17&0\\2&0&8&0&4&0&12&18&0&6&0&16&0&0&36\\0&3&0&5&0&12&0&0&15&0&4&0&16&27&0\\0&0&4&0&9&0&15&12&0&12&0&26&0&0&44\\0&8&0&12&0&38&0&0&48&0&20&0&60&92&0\\1&0&12&0&15&0&36&36&0&26&0&59&0&0&116\\4&0&18&0&12&0&36&52&0&24&0&60&0&0&132\\0&9&0&15&0&48&0&0&67&0&26&0&84&129&0\\0&0&6&0&12&0&26&24&0&24&0&48&0&0&96\\0&2&0&4&0&20&0&0&26&0&19&0&43&54&0\\1&0&16&0&26&0&59&60&0&48&0&119&0&0&224\\0&8&0&16&0&60&0&0&84&0&43&0&127&174&0\\0&17&0&27&0&92&0&0&129&0&54&0&174&269&0\\4&0&36&0&44&0&116&132&0&96&0&224&0&0&470\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&8&5&9&38&36&52&67&24&19&119&127&269&470&201&231&445&324&79\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.