\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 $\mathrm{ E }_{ 6 }$

Basic information
Coxeter number
$12$
dimension of group
$78$
exponents
$1,4,5,7,8,11$
Weyl group

$\mathrm{GO}_6^-(\mathbb{F}_2)\cong\mathrm{PSU}_4(\mathbb{F}_2):\mathbb{Z}/2\mathbb{Z}\cong\mathrm{SO}_5(\mathbb{F}_3)\cong\mathrm{Sp}_4(\mathbb{F}_3)$

order of the Weyl group
$51840=2^7\cdot 3^4\cdot 5$
Description of the root system
root space
$V\subseteq\mathbb{R}^8$ given by $v_6=v_7=-v_8$ for $\displaystyle\sum_{i=1}^8v_i\epsilon_i\in V$
roots
$\pm\epsilon_i\pm\epsilon_j$ for $1\leq i<j\leq 5$

$\displaystyle\pm\frac{1}{2}\left( \epsilon_8-\epsilon_7-\epsilon_6+\sum_{i=1}^5(-1)^{\nu(i)}\epsilon_i \right)$ for $\sum_{i=1}^5\nu(i)$ even

number of roots
$72$
simple roots
\begin{align} \alpha_1&=\frac{1}{2}(\epsilon_1+\epsilon_8)-\frac{1}{2}(\epsilon_2+\epsilon_3+\epsilon_4+\epsilon_5+\epsilon_6+\epsilon_7) \\ \alpha_2&=\epsilon_1+\epsilon_2 \\ \alpha_3&=\epsilon_2-\epsilon_1 \\ \alpha_4&=\epsilon_3-\epsilon_2 \\ \alpha_5&=\epsilon_4-\epsilon_3 \\ \alpha_6&=\epsilon_5-\epsilon_4 \end{align}
positive roots
\begin{array}{cc} \pm\epsilon_i+\epsilon_j & 1\leq i<j\leq 5 \\ \displaystyle\frac{1}{2}\left( \epsilon_8-\epsilon_7-\epsilon_6+\sum_{i=1}^5(-1)^{\nu(i)}\epsilon_i \right) & \sum_{i=1}^5\nu(i)\text{ even} \end{array}
highest root
\begin{align} \widetilde{\alpha}&=\frac{1}{2}(\epsilon_1+\epsilon_2+\epsilon_3+\epsilon_4+\epsilon_5-\epsilon_6-\epsilon_7+\epsilon_8) \\ &=\alpha_1+2\alpha_2+2\alpha_3+3\alpha_4+2\alpha_5+\alpha_6 \\ &=\omega_2 \end{align}
fundamental weights
\begin{align} \omega_1&=\frac{2}{3}(\epsilon_8-\epsilon_7-\epsilon_6) \\ &=\frac{1}{3}(4\alpha_1+3\alpha_2+5\alpha_3+6\alpha_4+4\alpha_5+2\alpha_6) \\ \omega_2&=\frac{1}{2}(\epsilon_1+\epsilon_2+\epsilon_3+\epsilon_4+\epsilon_5-\epsilon_6-\epsilon_7+\epsilon_8) \\ &=\alpha_1+2\alpha_2+2\alpha_3+3\alpha_4+2\alpha_5+\alpha_6 \\ \omega_3&=\frac{5}{6}(\epsilon_8-\epsilon_7-\epsilon_6)+\frac{1}{2}(-\epsilon_1+\epsilon_2+\epsilon_3+\epsilon_4+\epsilon_5) \\ &=\frac{1}{3}(5\alpha_1+6\alpha_2+10\alpha_3+12\alpha_4+8\alpha_5+4\alpha_6) \\ \omega_4&=\epsilon_3+\epsilon_4+\epsilon_5-\epsilon_6-\epsilon_7+\epsilon_8 \\ &=2\alpha_1+3\alpha_2+4\alpha_3+6\alpha_4+4\alpha_5+2\alpha_6 \\ \omega_5&=\frac{2}{3}(\epsilon_8-\epsilon_7-\epsilon_6)+\epsilon_4+\epsilon_5 \\ &=\frac{1}{3}(4\alpha_1+6\alpha_2+8\alpha_3+12\alpha_4+10\alpha_5+5\alpha_6) \\ \omega_6&=\frac{1}{3}(\epsilon_8-\epsilon_7-\epsilon_6)+\epsilon_5 \\ &=\frac{1}{3}(2\alpha_1+3\alpha_2+4\alpha_3+6\alpha_4+5\alpha_5+4\alpha_6) \end{align}
sum of positive roots
\begin{align} 2\rho&=2(\epsilon_2+2\epsilon_3+3\epsilon_4+4\epsilon_5+4(\epsilon_8-\epsilon_7-\epsilon_6)) \\ &=2(8\alpha_1+11\alpha_2+15\alpha_3+21\alpha_4+15\alpha_5+8\alpha_6) \end{align}
Cartan matrix
Cartan matrix
\begin{pmatrix} 2 & 0 & -1 & 0 & 0 & 0 \\ 0 & 2 & 0 & -1 & 0 & 0 \\ -1 & 0 & 2 & -1 & 0 & 0 \\ 0 & -1 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 \\ 0 & 0 & 0 & 0 & -1 & 2 \\ \end{pmatrix}
determinant
3