Name: | $\mathrm{U}(1)\times\mathrm{SU}(2)^2$ |
$\mathbb{R}$-dimension: | $7$ |
Description: | $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),B,C\in\mathrm{SU}(2)\right\}$ |
Symplectic form: | $\begin{bmatrix}J_2&0&0\\0&J_2&0\\0&0&J_2\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ |
Hodge circle: | $u\mapsto\mathrm{diag}(u,\bar u, u, \bar u, u, \bar u)$ |
Name: | $C_2$ |
Order: | $2$ |
Abelian: | yes |
Generators: | $\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$20$ |
$0$ |
$285$ |
$0$ |
$4949$ |
$0$ |
$97608$ |
$0$ |
$2104542$ |
$a_2$ |
$1$ |
$2$ |
$8$ |
$44$ |
$308$ |
$2507$ |
$22530$ |
$216967$ |
$2199888$ |
$23221661$ |
$253228082$ |
$2836717798$ |
$32505982085$ |
$a_3$ |
$1$ |
$0$ |
$9$ |
$0$ |
$692$ |
$0$ |
$90730$ |
$0$ |
$15598940$ |
$0$ |
$3162201336$ |
$0$ |
$716190153888$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$8$ |
$4$ |
$11$ |
$20$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$9$ |
$44$ |
$25$ |
$75$ |
$46$ |
$144$ |
$285$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$59$ |
$308$ |
$182$ |
$111$ |
$586$ |
$355$ |
$1174$ |
$710$ |
$2395$ |
$4949$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$453$ |
$2507$ |
$274$ |
$1491$ |
$897$ |
$5056$ |
$3024$ |
$1816$ |
$10442$ |
$6230$ |
$21817$ |
$12978$ |
$45983$ |
$97608$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$692$ |
$3863$ |
$22530$ |
$2310$ |
$13328$ |
$7925$ |
$46986$ |
$4728$ |
$27787$ |
$16480$ |
$99213$ |
$58520$ |
$34609$ |
$211131$ |
$$ |
$124201$ |
$452109$ |
$265272$ |
$973224$ |
$2104542$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&1&2&0&0&0&1&0&0&2\\0&2&0&2&0&5&0&0&5&0&2&0&5&8&0\\1&0&5&0&3&0&7&9&0&4&0&10&0&0&18\\0&2&0&3&0&7&0&0&8&0&3&0&9&13&0\\0&0&3&0&6&0&9&7&0&6&0&14&0&0&22\\0&5&0&7&0&22&0&0&25&0&12&0&32&45&0\\1&0&7&0&9&0&20&19&0&13&0&31&0&0&56\\2&0&9&0&7&0&19&25&0&12&0&32&0&0&62\\0&5&0&8&0&25&0&0&33&0&14&0&42&60&0\\0&0&4&0&6&0&13&12&0&12&0&23&0&0&44\\0&2&0&3&0&12&0&0&14&0&10&0&22&28&0\\1&0&10&0&14&0&31&32&0&23&0&61&0&0&106\\0&5&0&9&0&32&0&0&42&0&22&0&62&83&0\\0&8&0&13&0&45&0&0&60&0&28&0&83&122&0\\2&0&18&0&22&0&56&62&0&44&0&106&0&0&210\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&3&6&22&20&25&33&12&10&61&62&122&210&87&91&179&130&31\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=0$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_3=0$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=a_3=0$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|