Properties

Label 1.6.J.2.1c
  
Name \(J(J(E_1),E_1)\)
Weight $1$
Degree $6$
Real dimension $4$
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:$4$
Components:$2$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)\times\mathrm{SU}(2)_2$
$\mathbb{R}$-dimension:$4$
Description:$\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&\overline{B}\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),B\in\mathrm{SU}(2)\right\}$ 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:$\mathrm{diag}(u,\bar u, u,\bar u,\bar u,u)$

Component group

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

Subgroups and supergroups

Maximal subgroups:$J_1(E_1)$
Minimal supergroups:$J(J(E_2),J(E_1))$, $J(J(E_2),E_2)$${}^{\times 2}$, $J(J(E_3),E_3)$, $J_2(J(E_1))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $43$ $0$ $830$ $0$ $18627$ $0$ $458010$ $0$ $11976030$
$a_2$ $1$ $3$ $15$ $105$ $923$ $9233$ $99641$ $1129775$ $13271619$ $160178265$ $1975274093$ $24790665419$ $315730935685$
$a_3$ $1$ $0$ $18$ $0$ $2338$ $0$ $456520$ $0$ $107761906$ $0$ $28401505224$ $0$ $8045063991624$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $15$ $7$ $22$ $43$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $18$ $105$ $57$ $193$ $113$ $396$ $830$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $151$ $923$ $524$ $306$ $1881$ $1091$ $3995$ $2310$ $8588$ $18627$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1446$ $9233$ $839$ $5282$ $3049$ $19733$ $11332$ $6524$ $42887$ $24562$ $93918$ $53641$ $206892$ $458010$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $2338$ $15005$ $99641$ $8625$ $56787$ $32482$ $218300$ $18618$ $124318$ $70926$ $482145$ $273928$ $155894$ $1070262$
$$ $606711$ $2385675$ $1349586$ $5336820$ $11976030$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&2&5&0&1&0&3&0&0&8\\0&3&0&4&0&12&0&0&16&0&6&0&19&30&0\\2&0&10&0&7&0&21&28&0&15&0&36&0&0&80\\0&4&0&7&0&20&0&0&28&0&11&0&37&55&0\\0&0&7&0&14&0&28&25&0&25&0&52&0&0&104\\0&12&0&20&0&70&0&0&100&0&44&0&140&208&0\\2&0&21&0&28&0&71&75&0&60&0&135&0&0&288\\5&0&28&0&25&0&75&100&0&62&0&150&0&0&336\\0&16&0&28&0&100&0&0&151&0&66&0&215&322&0\\1&0&15&0&25&0&60&62&0&58&0&122&0&0&264\\0&6&0&11&0&44&0&0&66&0&34&0&101&146&0\\3&0&36&0&52&0&135&150&0&122&0&287&0&0&616\\0&19&0&37&0&140&0&0&215&0&101&0&328&480&0\\0&30&0&55&0&208&0&0&322&0&146&0&480&725&0\\8&0&80&0&104&0&288&336&0&264&0&616&0&0&1376\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&10&7&14&70&71&100&151&58&34&287&328&725&1376&631&646&1440&1149&295\end{bmatrix}$

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$