Properties

 Label 1.4.F.12.3a Name $$T$$ Weight $1$ Degree $4$ Real dimension $1$ Components $12$ Contained in $$\mathrm{USp}(4)$$ Identity component $$\mathrm{U}(1)_2$$ Component group $$A_4$$

Invariants

 Weight: $1$ Degree: $4$ $\mathbb{R}$-dimension: $1$ Components: $12$ Contained in: $\mathrm{USp}(4)$ Rational: yes

Identity component

 Name: $\mathrm{U}(1)_2$ $\mathbb{R}$-dimension: $1$ Description: $\left\{\begin{bmatrix}\alpha I_2&0\\0&\bar\alpha I_2\end{bmatrix}: \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ Symplectic form: $\begin{bmatrix}0&I_2\\-I_2&0\end{bmatrix}$ Hodge circle: $u\mapsto\mathrm{diag}(u, u,\bar u,\bar u)$

Component group

 Name: $A_4$ Order: $12$ Abelian: no Generators: $\begin{bmatrix}\frac{i+1}{2}&\frac{i+1}{2}&0&0\\\frac{i-1}{2}&\frac{-i+1}{2}&0&0\\0&0&\frac{-i+1}{2}&\frac{-i+1}{2}\\0&0&\frac{-i-1}{2}&\frac{i+1}{2}\end{bmatrix}, \begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $12$ $0$ $120$ $0$ $1540$ $0$ $21672$ $0$ $316008$
$a_2$ $1$ $1$ $4$ $12$ $52$ $236$ $1202$ $6378$ $35044$ $195924$ $1108834$ $6323978$ $36271314$

Moment simplex

 $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $1$ $2$ $4$ $6$ $12$ $12$ $24$ $52$ $120$ $52$ $110$ $256$ $620$ $1540$ $236$ $552$ $1344$ $3352$ $8484$ $21672$ $1202$ $2924$ $7316$ $18564$ $47516$ $122304$ $316008$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&1&0&1&0&0&0&1\\0&2&0&0&2&0&2&0&4&0\\0&0&3&1&0&1&0&5&0&5\\1&0&1&4&0&4&0&7&0&6\\0&2&0&0&8&0&6&0&14&0\\1&0&1&4&0&8&0&9&0&6\\0&2&0&0&6&0&10&0&16&0\\0&0&5&7&0&9&0&27&0&21\\0&4&0&0&14&0&16&0&32&0\\1&0&5&6&0&6&0&21&0&22\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&3&4&8&8&10&27&32&22\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-2$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$
$-$$1$$0$$0$$0$$0$$0$$0 a_1=0$$1/4$$0$$0$$0$$0$$0$$0$