Properties

Label 1.6.L.4.1c
  
Name \(L(C_4,C_2)\)
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 \(C_4\)

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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:$C_4$
Order:$4$
Abelian:yes
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 &0 & 0 \\0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$L_1(C_2)$
Minimal supergroups:$L_2(C_4)$, $L(J(C_4),J(C_2))$, $L(J(C_4),C_{4,1})$, $L(D_4,D_2)$, $L(D_{4,2},D_{2,1})$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $5$ $0$ $63$ $0$ $1310$ $0$ $34195$ $0$ $982170$ $0$ $29575854$
$a_2$ $1$ $3$ $17$ $135$ $1367$ $15773$ $194891$ $2501621$ $32882763$ $439414641$ $5945118347$ $81228148301$ $1118773707425$
$a_3$ $1$ $0$ $26$ $0$ $3762$ $0$ $971720$ $0$ $292656658$ $0$ $94097303496$ $0$ $31472833312008$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $5$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $17$ $10$ $31$ $63$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $26$ $135$ $80$ $279$ $162$ $597$ $1310$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $214$ $1367$ $782$ $454$ $3001$ $1708$ $6693$ $3780$ $15070$ $34195$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $2256$ $15773$ $1284$ $8862$ $4998$ $35663$ $19966$ $11220$ $81195$ $45320$ $185750$ $103390$ $426489$ $982170$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $3762$ $26576$ $194891$ $14898$ $108350$ $60368$ $447189$ $33700$ $248094$ $137876$ $1029629$ $570168$ $316254$ $2376862$
$$ $1314152$ $5498549$ $3035844$ $12742674$ $29575854$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&2&0&3&4&0&4&0&5&0&0&12\\0&5&0&5&0&16&0&0&23&0&11&0&29&47&0\\2&0&12&0&12&0&32&34&0&26&0&62&0&0&140\\0&5&0&11&0&28&0&0&41&0&17&0&69&93&0\\2&0&12&0&18&0&38&48&0&40&0&88&0&0&196\\0&16&0&28&0&108&0&0&168&0&80&0&264&404&0\\3&0&32&0&38&0&117&128&0&102&0&253&0&0&588\\4&0&34&0&48&0&128&164&0&132&0&306&0&0&716\\0&23&0&41&0&168&0&0&285&0&131&0&441&687&0\\4&0&26&0&40&0&102&132&0&122&0&248&0&0&604\\0&11&0&17&0&80&0&0&131&0&68&0&204&325&0\\5&0&62&0&88&0&253&306&0&248&0&611&0&0&1420\\0&29&0&69&0&264&0&0&441&0&204&0&742&1107&0\\0&47&0&93&0&404&0&0&687&0&325&0&1107&1733&0\\12&0&140&0&196&0&588&716&0&604&0&1420&0&0&3424\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&5&12&11&18&108&117&164&285&122&68&611&742&1733&3424&1703&1722&4146&3519&1009\end{bmatrix}$

Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.