Properties

Label 1.6.F.4.2a
  
Name \(F_{a,t}\)
Weight $1$
Degree $6$
Real dimension $7$
Components $4$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{SU}(2)^2\)
Component group \(C_2^2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$7$
Components:$4$
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:$C_2^2$
Order:$4$
Abelian:yes
Generators:$\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $14$ $0$ $165$ $0$ $2639$ $0$ $50253$ $0$ $1066659$
$a_2$ $1$ $2$ $7$ $32$ $191$ $1402$ $11905$ $111414$ $1114062$ $11681759$ $126982850$ $1420331497$ $16263790724$
$a_3$ $1$ $0$ $7$ $0$ $389$ $0$ $46510$ $0$ $7837347$ $0$ $1582522578$ $0$ $358153094508$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $7$ $3$ $8$ $14$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $7$ $32$ $17$ $47$ $28$ $85$ $165$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $38$ $191$ $109$ $67$ $333$ $202$ $645$ $390$ $1290$ $2639$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $262$ $1402$ $157$ $824$ $496$ $2710$ $1625$ $983$ $5499$ $3295$ $11368$ $6783$ $23786$ $50253$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $389$ $2087$ $11905$ $1250$ $7025$ $4187$ $24365$ $2504$ $14445$ $8595$ $51005$ $30169$ $17910$ $107950$
$$ $63672$ $230279$ $135408$ $494298$ $1066659$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&0&0&0&0&0&1\\0&2&0&1&0&3&0&0&3&0&1&0&1&5&0\\1&0&4&0&1&0&4&6&0&1&0&4&0&0&9\\0&1&0&3&0&4&0&0&4&0&0&0&6&6&0\\0&0&1&0&5&0&4&3&0&5&0&8&0&0&11\\0&3&0&4&0&12&0&0&13&0&6&0&16&23&0\\0&0&4&0&4&0&13&8&0&5&0&16&0&0&28\\2&0&6&0&3&0&8&17&0&6&0&14&0&0&31\\0&3&0&4&0&13&0&0&18&0&7&0&19&31&0\\0&0&1&0&5&0&5&6&0&10&0&12&0&0&22\\0&1&0&0&0&6&0&0&7&0&8&0&10&14&0\\0&0&4&0&8&0&16&14&0&12&0&33&0&0&53\\0&1&0&6&0&16&0&0&19&0&10&0&38&38&0\\0&5&0&6&0&23&0&0&31&0&14&0&38&64&0\\1&0&9&0&11&0&28&31&0&22&0&53&0&0&105\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&3&5&12&13&17&18&10&8&33&38&64&105&51&54&97&71&24\end{bmatrix}$

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$$0$
$a_1=0$$1/4$$0$$0$$0$$0$$0$$0$
$a_3=0$$1/4$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$1/4$$0$$0$$0$$0$$0$$0$