\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 $\mathrm{ B }_{ 3 }$

Basic information
Coxeter number
$2n=6$
dimension of group
$2n^2+n=21$
exponents
$1,3,5,\ldots,2n-1=1,3,5$
Weyl group

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

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

order of the Weyl group
$n!2^n=3!2^{ 3 }=48$
Description of the root system
root space
$V=\mathbb{R}^n$
roots
$\pm\epsilon_i\pm\epsilon_j,\pm\epsilon_i$
number of roots
$2n^2=18$
simple roots
$\alpha_i=\begin{cases}\epsilon_i-\epsilon_{i+1} & i<n \\ \epsilon_n & i=n\end{cases}$
positive roots
\begin{cases} \displaystyle\epsilon_i=\sum_{i\leq k\leq n}\alpha_k & 1\leq i\leq n \\ \displaystyle\epsilon_i-\epsilon_j=\sum_{i\leq 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\leq n}\alpha_k & 1\leq i<j\leq n \end{cases}
highest root
\begin{align} \widetilde{\alpha}&=\epsilon_1+\epsilon_2 \\ &=\alpha_1+2\alpha_2+\ldots+2\alpha_n \\ &=\begin{cases} 2\omega_2 & n=2 \\ \omega_2 & n\neq 3 \end{cases} \end{align}
fundamental weights
\begin{align} \omega_i&=\epsilon_1+\epsilon_2+\ldots+\epsilon_i & 1\leq i<n \\ &=\alpha_1+2\alpha_2+\ldots+(i-1)\alpha_{i-1}+i(\alpha_i+\ldots+\alpha_n) \\ \omega_n&=\frac{1}{2}(\epsilon_1+\ldots+\epsilon_n) \\ &=\frac{1}{2}(\alpha_1+2\alpha_2+\ldots+2\alpha_n) \end{align}
sum of positive roots
\begin{align} 2\rho&=(2n-1)\epsilon_1+(2n-3)\epsilon_2+\ldots+3\epsilon_{n-1}+\epsilon_n \\ &=(2n-1)\alpha_1+2(2n-2)\alpha_2+\ldots+i(2n-i)\alpha_i+\ldots+n^2\alpha_n \end{align}
Cartan matrix
Cartan matrix
\begin{pmatrix} 2 & -1 & 0 \\ -1 & 2 & -2 \\ 0 & -1 & 2 \\ \end{pmatrix}
determinant
$2$