Properties

 Label 1.6.N.12.4d Name $$J(A(3,2))$$ Weight $1$ Degree $6$ Real dimension $1$ Components $12$ Contained in $$\mathrm{USp}(6)$$ Identity component $$\mathrm{U}(1)_3$$ Component group $$D_6$$

Invariants

 Weight: $1$ Degree: $6$ $\mathbb{R}$-dimension: $1$ Components: $12$ Contained in: $\mathrm{USp}(6)$ Rational: yes

Identity component

 Name: $\mathrm{U}(1)_3$ $\mathbb{R}$-dimension: $1$ Description: $\left\{\begin{bmatrix}\alpha I_3&0,\\ 0&\bar \alpha I_3\end{bmatrix}: \alpha\bar\alpha=1,\ \alpha\in\mathbb{C}\right\}$ Symplectic form: $\begin{bmatrix} 0 & I_3\\ -I_3 & 0\end{bmatrix}$ Hodge circle: $u\mapsto \mathrm{diag}(u,u, u, \bar u, \bar u, \bar u)$

Component group

 Name: $D_6$ Order: $12$ Abelian: no Generators: $\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0& 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$