Properties

Label 1.6.L.4.2a
  
Name \(L(D_2,C_2)\)
Weight $1$
Degree $6$
Real dimension $6$
Components $4$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{U}(1)_2\)
Component group \(C_2^2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$6$
Components:$4$
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:$C_2^2$
Order:$4$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 &0 & 0 \\0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0& 0 & 0 & -1 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$L(C_2,C_1)$${}^{\times 2}$, $L_1(C_2)$
Minimal supergroups:$L(J(D_2),J(C_2))$, $L_2(D_2)$${}^{\times 3}$, $L(D_4,D_2)$, $L(D_{4,1},D_{2,1})$, $L(D_4,C_4)$${}^{\times 2}$, $L(D_{4,1},C_{4,1})$, $L(J(D_2),D_{2,1})$, $L(D_6,C_6)$, $L(D_6,D_3)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $51$ $0$ $1230$ $0$ $33635$ $0$ $978138$ $0$ $29546286$
$a_2$ $1$ $2$ $13$ $116$ $1279$ $15362$ $192967$ $2492576$ $32840059$ $439212158$ $5944154383$ $81223542116$ $1118751623689$
$a_3$ $1$ $0$ $18$ $0$ $3570$ $0$ $966600$ $0$ $292513298$ $0$ $94093174728$ $0$ $31472712201480$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $13$ $6$ $23$ $51$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $18$ $116$ $64$ $245$ $138$ $545$ $1230$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $182$ $1279$ $714$ $406$ $2851$ $1604$ $6461$ $3620$ $14710$ $33635$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $2120$ $15362$ $1188$ $8562$ $4790$ $34985$ $19502$ $10900$ $80139$ $44600$ $184102$ $102270$ $423913$ $978138$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $3570$ $25976$ $192967$ $14482$ $106994$ $59440$ $444073$ $33060$ $245982$ $136436$ $1024753$ $566872$ $314014$ $2369222$
$$ $1309000$ $5486565$ $3027780$ $12723858$ $29546286$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&2&5&0&2&0&5&0&0&12\\0&3&0&3&0&14&0&0&21&0&11&0&29&49&0\\1&0&10&0&9&0&28&34&0&24&0&63&0&0&140\\0&3&0&9&0&26&0&0&39&0&17&0&69&95&0\\1&0&9&0&18&0&37&45&0&40&0&89&0&0&196\\0&14&0&26&0&104&0&0&166&0&80&0&264&404&0\\2&0&28&0&37&0&115&126&0&100&0&254&0&0&588\\5&0&34&0&45&0&126&166&0&130&0&303&0&0&716\\0&21&0&39&0&166&0&0&283&0&131&0&441&689&0\\2&0&24&0&40&0&100&130&0&122&0&252&0&0&604\\0&11&0&17&0&80&0&0&131&0&68&0&204&325&0\\5&0&63&0&89&0&254&303&0&252&0&611&0&0&1420\\0&29&0&69&0&264&0&0&441&0&204&0&742&1107&0\\0&49&0&95&0&404&0&0&689&0&325&0&1107&1727&0\\12&0&140&0&196&0&588&716&0&604&0&1420&0&0&3424\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&10&9&18&104&115&166&283&122&68&611&742&1727&3424&1697&1726&4146&3517&1003\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$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$