\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 $\mathrm{ F }_{ 4 }$

Basic information
Coxeter number
$12$
dimension of group
$52$
exponents
$1,5,7,11$
Weyl group

$\mathrm{S}_3\rtimes(\mathrm{S}_4\rtimes(\mathbb{Z}/2\mathbb{Z}^{\oplus3})$

order of the Weyl group
$1152=2^7\cdot 3^2$
Description of the root system
root space
$V=\mathbb{R}^4$
roots
$\pm\epsilon_i$ for $1\leq i\leq 4$

$\pm\epsilon_i\pm\epsilon_j$ for $1\leq i<j\leq 4$

$\displaystyle\frac{1}{2}(\pm\epsilon_1+\pm\epsilon_2+\pm\epsilon_3+\pm\epsilon_4)$

number of roots
$48$
simple roots
\begin{align} \alpha_1&=\epsilon_2-\epsilon_3 \\ \alpha_2&=\epsilon_3-\epsilon_4 \\ \alpha_3&=\epsilon_4 \\ \alpha_4&=\frac{1}{2}(\epsilon_1-\epsilon_2-\epsilon_3-\epsilon_4) \end{align}
positive roots
\begin{array}{cc} \epsilon_i & 1\leq i\leq 4 \\ \epsilon_i\pm\epsilon_j & 1\leq i<j\leq 4 \\ \frac{1}{2}(\epsilon_1\pm\epsilon_2\pm\epsilon_3\pm\epsilon_4) \end{array}
highest root
\begin{align} \widetilde{\alpha}&=\epsilon_1+\epsilon_2 \\ &=2\alpha_1+3\alpha_3+4\alpha_3+2\alpha_4 \\ &=\omega_1 \end{align}
fundamental weights
\begin{align} \omega_1&=\epsilon_1+\epsilon_2 \\ &=2\alpha_1+3\alpha_2+4\alpha_3+2\alpha_4 \\ \omega_2&=2\epsilon_1+\epsilon_2+\epsilon_3 \\ &=3\alpha_1+6\alpha_2+8\alpha_3+4\alpha_4 \\ \omega_3&=\frac{1}{2}(3\epsilon_1+\epsilon_2+\epsilon_3+\epsilon_4) \\ &=2\alpha_1+4\alpha_2+6\alpha_3+3\alpha_4 \\ \omega_4&=\epsilon_1 \\ &=\alpha_1+2\alpha_2+3\alpha_2+2\alpha_4 \end{align}
sum of positive roots
\begin{align} 2\rho&=11\epsilon_1+5\epsilon_2+3\epsilon_3+\epsilon_4 \\ &=16\alpha_1+30\alpha_2+42\alpha_3+22\alpha_4 \end{align}
Cartan matrix
Cartan matrix
\begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 2 & -2 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \\ \end{pmatrix}
determinant
$1$