Properties

Label 1.6.F.2.1a
  
Name \(F_{at}\)
Weight $1$
Degree $6$
Real dimension $7$
Components $2$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{SU}(2)^2\)
Component group \(C_2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$7$
Components:$2$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)\times\mathrm{SU}(2)^2$
$\mathbb{R}$-dimension:$7$
Description:$\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),B,C\in\mathrm{SU}(2)\right\}$ Symplectic form:$\begin{bmatrix}J_2&0&0\\0&J_2&0\\0&0&J_2\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$
Hodge circle:$u\mapsto\mathrm{diag}(u,\bar u, u, \bar u, u, \bar u)$

Component group

Name:$C_2$
Order:$2$
Abelian:yes
Generators:$\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$F$
Minimal supergroups:$F_{a,t}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $20$ $0$ $285$ $0$ $4949$ $0$ $97608$ $0$ $2104542$
$a_2$ $1$ $2$ $8$ $44$ $308$ $2507$ $22530$ $216967$ $2199888$ $23221661$ $253228082$ $2836717798$ $32505982085$
$a_3$ $1$ $0$ $9$ $0$ $692$ $0$ $90730$ $0$ $15598940$ $0$ $3162201336$ $0$ $716190153888$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $8$ $4$ $11$ $20$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $9$ $44$ $25$ $75$ $46$ $144$ $285$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $59$ $308$ $182$ $111$ $586$ $355$ $1174$ $710$ $2395$ $4949$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $453$ $2507$ $274$ $1491$ $897$ $5056$ $3024$ $1816$ $10442$ $6230$ $21817$ $12978$ $45983$ $97608$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $692$ $3863$ $22530$ $2310$ $13328$ $7925$ $46986$ $4728$ $27787$ $16480$ $99213$ $58520$ $34609$ $211131$
$$ $124201$ $452109$ $265272$ $973224$ $2104542$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&1&2&0&0&0&1&0&0&2\\0&2&0&2&0&5&0&0&5&0&2&0&5&8&0\\1&0&5&0&3&0&7&9&0&4&0&10&0&0&18\\0&2&0&3&0&7&0&0&8&0&3&0&9&13&0\\0&0&3&0&6&0&9&7&0&6&0&14&0&0&22\\0&5&0&7&0&22&0&0&25&0&12&0&32&45&0\\1&0&7&0&9&0&20&19&0&13&0&31&0&0&56\\2&0&9&0&7&0&19&25&0&12&0&32&0&0&62\\0&5&0&8&0&25&0&0&33&0&14&0&42&60&0\\0&0&4&0&6&0&13&12&0&12&0&23&0&0&44\\0&2&0&3&0&12&0&0&14&0&10&0&22&28&0\\1&0&10&0&14&0&31&32&0&23&0&61&0&0&106\\0&5&0&9&0&32&0&0&42&0&22&0&62&83&0\\0&8&0&13&0&45&0&0&60&0&28&0&83&122&0\\2&0&18&0&22&0&56&62&0&44&0&106&0&0&210\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&3&6&22&20&25&33&12&10&61&62&122&210&87&91&179&130&31\end{bmatrix}$

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$