\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 $\mathrm{ E }_{ 8 }$

Basic information
Coxeter number
$30$
dimension of group
$248$
exponents
$1,7,11,13,17,19,23,29$
Weyl group

$2\cdot\mathrm{GO}_8^+(\mathbb{F}_2)$

order of the Weyl group
$696729600=2^{14}\cdot 3^5\cdot 5^2\cdot 7$
Description of the root system
root space
$V=\mathbb{R}^8$
roots
$\pm\epsilon_i\pm\epsilon_j$ for $1\leq i< j\leq 8$, $\displaystyle\frac{1}{2}\sum_{i=1}^8(-1)^{\nu(i)}$ for $\displaystyle\sum_{i=1}^8\nu(i)$ even
number of roots
$240$
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 \\ \alpha_8&=\epsilon_7-\epsilon_6 \end{align}
positive roots
\begin{array}{cc} \pm\epsilon_i+\epsilon_j & 1\leq i<j\leq 8 \\ \displaystyle\frac{1}{2}\left( \epsilon_8+\sum_{i=1}^7(-1)^{\nu(i)}\epsilon_i \right) & \sum_{i=1}^7\nu(i)\text{ even} \end{array}
highest root
\begin{align} \widetilde{\alpha}&=\epsilon_7+\epsilon_8 \\ &=2\alpha_1+3\alpha_2+4\alpha_3+6\alpha_4+5\alpha_5+4\alpha_6+3\alpha_7+2\alpha_8 \\ &=\omega_8 \end{align}
fundamental weights
\begin{align} \omega_1&=2\epsilon_8 \\ &=4\alpha_1+5\alpha_2+7\alpha_3+10\alpha_4+8\alpha_5+6\alpha_6+4\alpha_7+2\alpha_8 \\ \omega_2&=\frac{1}{2}(\epsilon_1+\epsilon_2+\epsilon_3+\epsilon_4+\epsilon_5+\epsilon_6+\epsilon_7+5\epsilon_8) \\ &=5\alpha_1+8\alpha_2+10\alpha_3+15\alpha_4+12\alpha_5+9\alpha_6+6\alpha_7+3\alpha_8 \\ \omega_3&=\frac{1}{2}(-\epsilon_1+\epsilon_2+\epsilon_3+\epsilon_4+\epsilon_5+\epsilon_6+\epsilon_7+7\epsilon_8) \\ &=7\alpha_1+10\alpha_2+14\alpha_3+20\alpha_4+16\alpha_5+12\alpha_6+8\alpha_7+4\alpha_8 \\ \omega_4&=\epsilon_3+\epsilon_4+\epsilon_5+\epsilon_6+\epsilon_7+5\epsilon_8 \\ &=10\alpha_1+15\alpha_2+20\alpha_3+30\alpha_4+24\alpha_5+18\alpha_6+12\alpha_7+6\alpha_8 \\ \omega_5&=\epsilon_4+\epsilon_5+\epsilon_6+\epsilon_7+4\epsilon_8 \\ &=8\alpha_1+12\alpha_2+16\alpha_3+24\alpha_4+20\alpha_5+15\alpha_6+10\alpha_7+5\alpha_8 \\ \omega_6&=\epsilon_5+\epsilon_6+\epsilon_7+3\epsilon_8 \\ &=6\alpha_1+9\alpha_2+12\alpha_3+18\alpha_4+15\alpha_5+12\alpha_6+8\alpha_7+4\alpha_8 \\ \omega_7&=\epsilon_6+\epsilon_7+2\epsilon_8 \\ &=4\alpha_1+6\alpha_2+8\alpha_3+12\alpha_4+10\alpha_5+8\alpha_6+6\alpha_7+3\alpha_8 \\ \omega_8&=\epsilon_7+\epsilon_8 \\ &=5\alpha_1+8\alpha_2+10\alpha_3+15\alpha_4+12\alpha_5+9\alpha_6+6\alpha_7+3\alpha_8 \end{align}
sum of positive roots
\begin{align} 2\rho&=(2\epsilon_2+\epsilon_3+3\epsilon_4+4\epsilon_5+5\epsilon_6+6\epsilon_7+23\epsilon_8) \\ &=2(46\alpha_1+68\alpha_2+91\alpha_3+135\alpha_4+110\alpha_5+84\alpha_6+57\alpha_7+29\alpha_8) \end{align}
Cartan matrix
Cartan matrix
\begin{pmatrix} 2 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 2 & -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 \\ \end{pmatrix}
determinant
1