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Properties

Label 1.6.L.6.2c
  
Name \(L_1(J(C_3))\)
Weight $1$
Degree $6$
Real dimension $6$
Components $6$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{U}(1)_2\)
Component group \(C_6\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$6$
Components:$6$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

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:$u\mapsto\mathrm{diag}(u,\bar u, u, u,\bar u, \bar u)$

Component group

Name:$C_6$
Order:$6$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$L_1(J(C_1))$, $L_1(C_3)$
Minimal supergroups:$L_1(J(T))$, $L(J(D_3),J(C_3))$, $L_1(J(C_6))$, $L_1(J(D_3))$, $L_2(J(C_3))$, $L(J(C_6),J(C_3))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $48$ $0$ $960$ $0$ $24080$ $0$ $673344$ $0$ $19982424$
$a_2$ $1$ $2$ $12$ $95$ $964$ $10977$ $133657$ $1696550$ $22139292$ $294567017$ $3975487207$ $54241821240$ $746516688397$
$a_3$ $1$ $0$ $18$ $0$ $2622$ $0$ $660080$ $0$ $196186158$ $0$ $62823369168$ $0$ $20989719099888$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $12$ $6$ $22$ $48$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $18$ $95$ $56$ $200$ $114$ $432$ $960$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $146$ $964$ $550$ $322$ $2120$ $1212$ $4730$ $2680$ $10630$ $24080$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1576$ $10977$ $888$ $6186$ $3494$ $24752$ $13910$ $7860$ $56178$ $31500$ $128112$ $71610$ $293216$ $673344$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $2622$ $18384$ $133657$ $10338$ $74530$ $41660$ $305722$ $23308$ $170130$ $94852$ $701868$ $389812$ $216974$ $1615964$
$$ $895972$ $3729516$ $2064384$ $8624952$ $19982424$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&4&0&3&0&4&0&0&8\\0&4&0&2&0&12&0&0&16&0&10&0&18&36&0\\1&0&9&0&8&0&23&23&0&14&0&47&0&0&96\\0&2&0&10&0&20&0&0&24&0&10&0&56&64&0\\2&0&8&0&16&0&24&37&0&31&0&63&0&0&132\\0&12&0&20&0&78&0&0&116&0&58&0&182&278&0\\1&0&23&0&24&0&87&85&0&60&0&177&0&0&400\\4&0&23&0&37&0&85&120&0&101&0&207&0&0&484\\0&16&0&24&0&116&0&0&204&0&92&0&288&470&0\\3&0&14&0&31&0&60&101&0&101&0&164&0&0&408\\0&10&0&10&0&58&0&0&92&0&54&0&130&224&0\\4&0&47&0&63&0&177&207&0&164&0&420&0&0&952\\0&18&0&56&0&182&0&0&288&0&130&0&528&736&0\\0&36&0&64&0&278&0&0&470&0&224&0&736&1166&0\\8&0&96&0&132&0&400&484&0&408&0&952&0&0&2300\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&9&10&16&78&87&120&204&101&54&420&528&1166&2300&1172&1167&2844&2395&766\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$1/2$$1/6$$0$$0$$1/3$$0$
$a_1=0$$0$$0$$0$$0$$0$$0$$0$
$a_3=0$$0$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$0$$0$$0$$0$$0$$0$$0$