B3—grassmannian.info

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+n\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: $2$

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