\begin{equation} \DeclareMathOperator\Gr{Gr} \DeclareMathOperator\LGr{LGr} \DeclareMathOperator\OGr{OGr} \DeclareMathOperator\SGr{SGr} \DeclareMathOperator\Kzero{K_0} \DeclareMathOperator\index{i} \DeclareMathOperator\rk{rk} \end{equation}

Grassmannian.info

A periodic table of (generalised) Grassmannians.

Dynkin type $\mathrm{ G }_{ 2 }$

Basic information
Coxeter number
$6$
dimension of group
$14$
exponents
$1,5$
Weyl group

dihedral group of order 12

order of the Weyl group
$12$
Description of the root system
root space
$V\subseteq\mathbb{R}^{3}$ given by $v_1+v_2+v_3=0$ for $v_1\epsilon_1+v_2\epsilon_2+v_3\epsilon_3\in V$
roots
$\pm(\epsilon_1-\epsilon_2),\pm(\epsilon_1-\epsilon_3),\pm(\epsilon_2-\epsilon_3),\pm(2\epsilon_1-\epsilon_2-\epsilon_3),\pm(2\epsilon_2-\epsilon_1-\epsilon_3),\pm(2\epsilon_3-\epsilon_1-\epsilon_2)$
number of roots
$12$
simple roots
$\alpha_1=\epsilon_1-\epsilon_2, \alpha_2=-2\epsilon_1+\epsilon_2+\epsilon_3$
positive roots
$\alpha_1,\alpha_2,\alpha_1+\alpha_2,2\alpha_1+\alpha_2,3\alpha_1+\alpha_2,3\alpha_1+2\alpha_2$
highest root
\begin{align} \widetilde{\alpha}&=-\epsilon_1-\epsilon_2+2\epsilon_3 \\ &=3\alpha_1+2\alpha_2 \\ &=\omega_2 \end{align}
fundamental weights
\begin{align} \omega_1&=2\alpha_1+\alpha_2 \\ \omega_2&=3\alpha_1+2\alpha_2 \end{align}
sum of positive roots
$2\rho=2(5\alpha_1+3\alpha_2)$
Cartan matrix
Cartan matrix
\begin{pmatrix} 2 & -1 \\ -3 & 2 \\ \end{pmatrix}
determinant
$1$