\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 $\mathrm{ E }_{ 7 }$

Basic information
Coxeter number
$18$
dimension of group
$133$
exponents
$1,5,7,9,11,13,17$
Weyl group

$\mathrm{GO}_7(\mathbb{F}_2)\times 2\cong\mathrm{Sp}_6(\mathbb{F}_2)\times\mathbb{Z}/2\mathbb{Z}$

order of the Weyl group
$2903040=2^{10}\cdot 3^4\cdot 5\cdot 7$
Description of the root system
root space
$V\subseteq\mathbb{R}^8$ orthogonal to $\epsilon_7+\epsilon_8$
roots
$\pm\epsilon_i\pm\epsilon_j$ for $1\leq i< j\leq 6$

$\pm(\epsilon_7-\epsilon_8)$

$\displaystyle\pm\frac{1}{2}\left( \epsilon_7-\epsilon_8+\sum_{i=1}^6(-1)^{\nu(i)}\epsilon_i \right)$ for $\displaystyle\sum_{i=1}^6\nu(i)$ odd

number of roots
$126$
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 \\ \alpha_7&=\epsilon_6-\epsilon_5 \end{align}
positive roots
\begin{array}{cc} \pm\epsilon_i+\epsilon_j & 1\leq i<j\leq 6 \\ -\epsilon_7+\epsilon_8 \\ \displaystyle\frac{1}{2}(-\epsilon_7+\epsilon_8+\sum_{i=1}^6(-1)^{\nu(i)}\epsilon_i & \sum_{i=1}^6\nu(i) \text{ odd} \end{array}
highest root
\begin{align} \widetilde{\alpha}&=\epsilon_8-\epsilon_7 \\ &=2\alpha_1+2\alpha_2+3\alpha_3+4\alpha_4+3\alpha_5+2\alpha_6+\alpha_7 \\ &=\omega_1 \end{align}
fundamental weights
\begin{align} \omega_1&=\epsilon_8-\epsilon_7 \\ &=2\alpha_1+2\alpha_2+3\alpha_3+4\alpha_4+3\alpha_5+2\alpha_6+\alpha_7 \\ \omega_2&=\frac{1}{2}(\epsilon_1+\epsilon_2+\epsilon_3+\epsilon_4+\epsilon_5+\epsilon_6-2\epsilon_7+2\epsilon_8) \\ &=\frac{1}{2}(4\alpha_1+7\alpha_2+8\alpha_3+12\alpha_4+9\alpha_5+8\alpha_6+3\alpha_7) \\ \omega_3&=\frac{1}{2}(-\epsilon_1+\epsilon_2+\epsilon_3+\epsilon_4+\epsilon_5+\epsilon_6-3\epsilon_7+3\epsilon_8) \\ &=3\alpha_1+4\alpha_2+6\alpha_3+8\alpha_4+6\alpha_5+4\alpha_6+2\alpha_7 \\ \omega_4&=\epsilon_3+\epsilon_4+\epsilon_5+\epsilon_6+2(\epsilon_8-\epsilon_7) \\ &=4\alpha_1+6\alpha_2+8\alpha_3+12\alpha_4+9\alpha_5+6\alpha_6+3\alpha_7 \\ \omega_5&=\epsilon_4+\epsilon_5+\epsilon_6+\frac{3}{2}(\epsilon_8-\epsilon_7) \\ &=\frac{1}{2}(6\alpha_1+9\alpha_2+12\alpha_3+18\alpha_4+15\alpha_5+10\alpha_6+5\alpha_7) \\ \omega_6&=\epsilon_5+\epsilon_6-\epsilon_7+\epsilon_8 \\ &=2\alpha_1+3\alpha_2+4\alpha_3+6\alpha_4+5\alpha_5+4\alpha_6+2\alpha_7 \\ \omega_7&=\epsilon_6+\frac{1}{2}\epsilon_8-\epsilon_7) \\ &=\frac{1}{2}(2\alpha_1+3\alpha_2+4\alpha_3+6\alpha_4+5\alpha_5+4\alpha_6+3\alpha_7) \end{align}
sum of positive roots
\begin{align} 2\rho&=2\epsilon_2+4\epsilon_3+6\epsilon_4+8\epsilon_5+10\epsilon_6-17\epsilon_7+17\epsilon_8 \\ &=34\alpha_1+49\alpha_2+66\alpha_3+96\alpha_4+75\alpha_5+52\alpha_6+27\alpha_7 \end{align}
Cartan matrix
Cartan matrix
\begin{pmatrix} 2 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & -1 & 0 & 0 & 0 \\ -1 & 0 & 2 & -1 & 0 & 0 & 0 \\ 0 & -1 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 \\ \end{pmatrix}
determinant
2