\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 $\mathrm{ D }_{ 3 }$

Basic information
Coxeter number
$2n-2=4$
dimension of group
$2n^2-n=15$
exponents
$1,3,5,\ldots,2n-3,n-1=1,2,3$
Weyl group

$\mathrm{S}_{ 3 }\rtimes(\mathbb{Z}/2\mathbb{Z})^{\oplus 2 }$

$\mathrm{S}_n$ permutes the $\epsilon_i$, $(\mathbb{Z}/2\mathbb{Z})^{\oplus n-1}$ sends $\epsilon_i$ to $(\pm1)_i\epsilon_i$ and $\prod_{i=1}^n(\pm1)_i=1$

order of the Weyl group
$n!2^{n-1}=3!2^{ 2 }=24$
Description of the root system
root space
$V=\mathbb{R}^n$
roots
$\pm\epsilon_i\pm\epsilon_j$
number of roots
$2n(n-1)=12$
simple roots
$\alpha_i=\begin{cases}\epsilon_i-\epsilon_{i+1} & i<n \\ \epsilon_{n-1}+\epsilon_n & i=n\end{cases}$
positive roots
\begin{aligned} \displaystyle\epsilon_i-\epsilon_j&=\sum_{i<k<j}\alpha_k & 1\leq i<j\leq n \\ \displaystyle\epsilon_i+\epsilon_j&=\sum_{i\leq k<j}\alpha_k+2\sum_{j\leq k<n-1}\alpha_k+\alpha_{n-1}+\alpha_n & 1\leq i<j< n \\ \displaystyle\epsilon_i+\epsilon_n&=\sum_{i\leq k<n-2}\alpha_k & 1\leq i<n \end{aligned}
highest root
\begin{align} \widetilde{\alpha}&=\epsilon_1+\epsilon_2 \\ &=\alpha_1+2\alpha_2+\ldots+2\alpha_{n-2}+\alpha_{n-1}+\alpha_n \\ &=\begin{cases} \omega_2+\omega_3 & n=3 \\ \omega_2 & n\geq 4 \end{cases} \end{align}
fundamental weights
\begin{align} \omega_i&=\epsilon_1+\epsilon_2+\ldots+\epsilon_i & 1\leq i\leq n-2 \\ &=\alpha_1+2\alpha_2+\ldots+(i-1)\alpha_{i-1}+i(\alpha_i+\alpha_{i+1}+\ldots+\alpha_{n-2})+\frac{1}{2}i(\alpha_{n-1}+\alpha_n) \\ \omega_{n-1}&=\frac{1}{2}(\epsilon_1+\epsilon_2+\ldots+\epsilon_{n-2}+\epsilon_{n-1}-\epsilon_n) \\ &=\frac{1}{2}\left( \alpha_1+2\alpha_2+\ldots+(n-2)\alpha_{n-2}+\frac{1}{2}n\alpha_{n-1}+\frac{1}{2}(n-2)\alpha_n \right) \\ \omega_n&=\frac{1}{2}(\epsilon_1+\epsilon_2+\ldots+\epsilon_{n-2}+\epsilon_{n-1}-\epsilon_n) \\ &=\frac{1}{2}\left( \alpha_1+2\alpha_2+\ldots+(n-2)\alpha_{n-2}+\frac{1}{2}(n-2)\alpha_{n-1}+\frac{1}{2}n\alpha_n \right) \end{align}
sum of positive roots
\begin{align} 2\rho&=2(n-1)\epsilon_1+2(n-2)\epsilon_2+\ldots+2\epsilon_{n-1} \\ &=2(n-1)\alpha+2(2n-3)\alpha_2+\ldots+2\left( in-\frac{i(i+1)}{2} \right)\alpha_i+\ldots+\frac{n(n-1)}{2}(\alpha_{n-1}+\alpha_n) \end{align}
Cartan matrix
Cartan matrix
\begin{pmatrix} 2 & -1 & -1 \\ -1 & 2 & 0 \\ -1 & 0 & 2 \\ \end{pmatrix}
determinant
$4$