\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 $\mathrm{ C }_{ 2 }$

Basic information
Coxeter number
$2n=4$
dimension of group
$2n^2+n=10$
exponents
$1,3,5,\ldots,2n-1=1,3$
Weyl group

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

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

order of the Weyl group
$n!2^n=2!2^{ 2 }=8$
Description of the root system
root space
$V=\mathbb{R}^n$
roots
$\pm\epsilon_i\pm\epsilon_j,\pm2\epsilon_i$
number of roots
$2n^2=8$
simple roots
$\alpha_i=\begin{cases}\epsilon_i-\epsilon_{i+1} & i<n \\ 2\epsilon_n & i=n\end{cases}$
positive roots
\begin{cases} \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<n}\alpha_k+\alpha_n & 1\leq i<j\leq n \\ \displaystyle2\epsilon_i=2\sum_{i\leq k<n}\alpha_k+\alpha_n & 1\leq i\leq n \end{cases}
highest root
\begin{align} \widetilde{\alpha}&=2\epsilon_1 \\ &=2\alpha_1+2\alpha_2+\ldots+2\alpha_{n-1}+\alpha_n \\ &=2\omega_1 \end{align}
fundamental weights
\begin{align} \omega_i&=\epsilon_1+\epsilon_2+\ldots+\epsilon_i & 1\leq i\leq n \\ &=\alpha_1+2\alpha_2+\ldots+(i-1)\alpha_{i-1}+i\left( \alpha_i+\alpha_{i+1}+\ldots+\alpha_{n-1}+\frac{1}{2}\alpha_n \right) \end{align}
sum of positive roots
\begin{align} 2\rho&=2n\epsilon_1+(2n-2)\epsilon_2+\ldots+4\epsilon_{n-1}+2\epsilon_n \\ &=2n\alpha_1+2(2n-1)\alpha_2+\ldots+i(2n-i+1)\alpha_i+\ldots+(n-1)(n+2)\alpha_{n-1}+\frac{1}{2}n(n+1)\alpha_n \end{align}
Cartan matrix
Cartan matrix
\begin{pmatrix} 2 & -1 \\ -2 & 2 \\ \end{pmatrix}
determinant
$2$