Name: | $\mathrm{U}(1)\times\mathrm{U}(1)_2$ |
$\mathbb{R}$-dimension: | $6$ |
Description: | $\left\{\begin{bmatrix}A&0&0\\0&\alpha I_2&0\\0&0&\bar\alpha I_2\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),\ \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ |
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: | $u\mapsto\mathrm{diag}(u,\bar u, u, u,\bar u, \bar u)$ |
Name: | $C_2$ |
Order: | $2$ |
Abelian: | yes |
Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 &0 \\0 & 0 & i & 0 & 0 & 0 \\0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i \\\end{bmatrix}$ |
Maximal subgroups: | $L_1(C_1)$ |
Minimal supergroups: | $L_2(C_2)$, $L(D_3,C_3)$, $L(D_{2,1},C_{2,1})$, $L(C_6,C_3)$, $L(D_2,C_2)$${}^{\times 2}$, $L(J(C_2),C_{2,1})$, $L(J(C_2),J(C_1))$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$5$ |
$0$ |
$99$ |
$0$ |
$2450$ |
$0$ |
$67235$ |
$0$ |
$1956150$ |
$0$ |
$59092110$ |
$a_2$ |
$1$ |
$3$ |
$23$ |
$225$ |
$2539$ |
$30673$ |
$385793$ |
$4984759$ |
$65679011$ |
$878421177$ |
$11888299813$ |
$162447058579$ |
$2237503173589$ |
$a_3$ |
$1$ |
$0$ |
$34$ |
$0$ |
$7122$ |
$0$ |
$1933000$ |
$0$ |
$585024146$ |
$0$ |
$188186317704$ |
$0$ |
$62945423976072$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$5$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$23$ |
$12$ |
$45$ |
$99$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$34$ |
$225$ |
$126$ |
$487$ |
$276$ |
$1087$ |
$2450$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$362$ |
$2539$ |
$1424$ |
$806$ |
$5695$ |
$3202$ |
$12913$ |
$7240$ |
$29410$ |
$67235$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$4232$ |
$30673$ |
$2376$ |
$17112$ |
$9574$ |
$69951$ |
$38992$ |
$21780$ |
$160257$ |
$89180$ |
$368174$ |
$204540$ |
$847791$ |
$1956150$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$7122$ |
$51932$ |
$385793$ |
$28946$ |
$213956$ |
$118856$ |
$888095$ |
$66120$ |
$491928$ |
$272852$ |
$2049449$ |
$1133704$ |
$627958$ |
$4738374$ |
$$ |
$2617930$ |
$10973025$ |
$6055560$ |
$25447590$ |
$59092110$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&2&0&5&8&0&4&0&11&0&0&24\\0&5&0&7&0&28&0&0&41&0&21&0&61&97&0\\2&0&18&0&20&0&56&68&0&50&0&126&0&0&280\\0&7&0&15&0&52&0&0&81&0&37&0&131&193&0\\2&0&20&0&32&0&76&90&0&74&0&178&0&0&392\\0&28&0&52&0&208&0&0&332&0&160&0&528&808&0\\5&0&56&0&76&0&223&258&0&208&0&507&0&0&1176\\8&0&68&0&90&0&258&322&0&256&0&610&0&0&1432\\0&41&0&81&0&332&0&0&561&0&259&0&891&1375&0\\4&0&50&0&74&0&208&256&0&226&0&504&0&0&1208\\0&21&0&37&0&160&0&0&259&0&132&0&416&647&0\\11&0&126&0&178&0&507&610&0&504&0&1219&0&0&2840\\0&61&0&131&0&528&0&0&891&0&416&0&1462&2223&0\\0&97&0&193&0&808&0&0&1375&0&647&0&2223&3449&0\\24&0&280&0&392&0&1176&1432&0&1208&0&2840&0&0&6848\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&5&18&15&32&208&223&322&561&226&132&1219&1462&3449&6848&3367&3436&8250&7005&1953\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=0$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_3=0$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=a_3=0$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|