F4—grassmannian.info

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: $1$

Grassmannian.info

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