Properties

Label 1.6.F.2.1b
  
Name \(F_{t}\)
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:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 &1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $23$ $0$ $295$ $0$ $4984$ $0$ $97734$ $0$ $2105004$
$a_2$ $1$ $2$ $8$ $44$ $308$ $2507$ $22530$ $216967$ $2199888$ $23221661$ $253228082$ $2836717798$ $32505982085$
$a_3$ $1$ $0$ $10$ $0$ $698$ $0$ $90780$ $0$ $15599430$ $0$ $3162206628$ $0$ $716190214872$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $8$ $4$ $12$ $23$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $10$ $44$ $26$ $77$ $46$ $147$ $295$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $60$ $308$ $184$ $114$ $590$ $358$ $1180$ $710$ $2405$ $4984$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $456$ $2507$ $274$ $1496$ $900$ $5065$ $3030$ $1826$ $10454$ $6240$ $21837$ $12978$ $46018$ $97734$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $698$ $3870$ $22530$ $2316$ $13340$ $7934$ $47007$ $4728$ $27802$ $16490$ $99240$ $58540$ $34644$ $211171$
$$ $124236$ $452179$ $265272$ $973350$ $2105004$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&0&2&0&1&0&1&0&0&2\\0&3&0&1&0&5&0&0&5&0&3&0&3&9&0\\1&0&5&0&3&0&8&8&0&3&0&10&0&0&18\\0&1&0&5&0&7&0&0&7&0&1&0&13&11&0\\1&0&3&0&7&0&6&9&0&8&0&14&0&0&22\\0&5&0&7&0&22&0&0&25&0&12&0&32&45&0\\0&0&8&0&6&0&24&17&0&9&0&31&0&0&56\\2&0&8&0&9&0&17&26&0&16&0&32&0&0&62\\0&5&0&7&0&25&0&0&35&0&15&0&39&61&0\\1&0&3&0&8&0&9&16&0&17&0&22&0&0&44\\0&3&0&1&0&12&0&0&15&0&13&0&17&30&0\\1&0&10&0&14&0&31&32&0&22&0&61&0&0&106\\0&3&0&13&0&32&0&0&39&0&17&0&73&78&0\\0&9&0&11&0&45&0&0&61&0&30&0&78&125&0\\2&0&18&0&22&0&56&62&0&44&0&106&0&0&210\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&5&5&7&22&24&26&35&17&13&61&73&125&210&99&92&191&135&45\end{bmatrix}$

Event probabilities

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