Properties

Label 1.6.H.2.1c
  
Name \(H_{a}\)
Weight $1$
Degree $6$
Real dimension $3$
Components $2$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)^3\)
Component group \(C_2\)

Learn more

Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$3$
Components:$2$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)^3$
$\mathbb{R}$-dimension:$3$
Description:$\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}: A,B,C\in\mathrm{U}(1)\subseteq\mathrm{SU}(2)\right\}$ 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:$\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$H$
Minimal supergroups:$H_{a,b}$${}^{\times 2}$, $H_{a,bc}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $5$ $0$ $63$ $0$ $1130$ $0$ $24815$ $0$ $618030$ $0$ $16674966$
$a_2$ $1$ $3$ $17$ $123$ $1089$ $11043$ $122489$ $1440771$ $17627969$ $221716707$ $2845847457$ $37102986891$ $489809824425$
$a_3$ $1$ $0$ $24$ $0$ $2748$ $0$ $567360$ $0$ $147281260$ $0$ $42454185984$ $0$ $13001375180640$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $5$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $17$ $10$ $31$ $63$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $24$ $123$ $74$ $249$ $150$ $525$ $1130$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $184$ $1089$ $642$ $384$ $2331$ $1374$ $5083$ $2980$ $11190$ $24815$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1716$ $11043$ $1008$ $6390$ $3720$ $24393$ $14078$ $8168$ $54345$ $31260$ $121730$ $69790$ $273805$ $618030$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $2748$ $17964$ $122489$ $10356$ $69810$ $39940$ $275515$ $22936$ $156642$ $89352$ $622467$ $353020$ $200840$ $1410910$
$$ $798350$ $3206651$ $1810620$ $7304850$ $16674966$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&2&0&3&4&0&2&0&5&0&0&8\\0&5&0&5&0&16&0&0&17&0&11&0&23&35&0\\2&0&12&0&12&0&26&28&0&16&0&50&0&0&88\\0&5&0&9&0&24&0&0&29&0&15&0&47&63&0\\2&0&12&0&18&0&32&36&0&24&0&70&0&0&120\\0&16&0&24&0&88&0&0&112&0&64&0&176&256&0\\3&0&26&0&32&0&83&88&0&62&0&171&0&0&344\\4&0&28&0&36&0&88&108&0&76&0&196&0&0&408\\0&17&0&29&0&112&0&0&173&0&83&0&263&395&0\\2&0&16&0&24&0&62&76&0&68&0&142&0&0&328\\0&11&0&15&0&64&0&0&83&0&54&0&134&199&0\\5&0&50&0&70&0&171&196&0&142&0&393&0&0&792\\0&23&0&47&0&176&0&0&263&0&134&0&434&627&0\\0&35&0&63&0&256&0&0&395&0&199&0&627&953&0\\8&0&88&0&120&0&344&408&0&328&0&792&0&0&1800\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&5&12&9&18&88&83&108&173&68&54&393&434&953&1800&853&854&1986&1631&441\end{bmatrix}$

Event probabilities

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