A2—grassmannian.info

Basic information

Coxeter number

$n+1=3$

dimension of group

$n^2+2n=8$

exponents

$1,2,\ldots,n=1,2$

Weyl group

$\mathrm{S}_{ 3}$, the symmetric group on $n+1=3$ elements

permuting the $\epsilon_i$

order of the Weyl group

$(n+1)!=3!=6$

Description of the root system

root space: $V\subseteq\mathbb{R}^{n+1}$ given by $\displaystyle\sum_{i=1}^{n+1}v_i=0$ for $\displaystyle \sum_{i=1}^{n+1}v_i\epsilon_i\in V$
roots: $\epsilon_i-\epsilon_j$, for $i\neq j$ and $i,j=1,\ldots,n+1$
number of roots: $n(n+1)=6$
simple roots: $\alpha_i=\epsilon_i-\epsilon_{i+1}$
positive roots: $\displaystyle\epsilon_i-\epsilon_j=\sum_{i\leq k<j}\alpha_k$ for $1\leq i<j\leq n+1$
highest root: \begin{align} \widetilde{\alpha}&=\epsilon_1-\epsilon_{n+1} \\ &=\alpha_1+\ldots+\alpha_n \\ &=\omega_1+\omega_n \end{align}
fundamental weights: $\displaystyle\omega_i=(\epsilon_1+\ldots+\epsilon_i)-\frac{i}{n+1}\sum_{j=1}^{n+1}\epsilon_j$
sum of positive roots: \begin{align} 2\rho&=n\epsilon_1+(n-2)\epsilon_2+(n-4)\epsilon_3-(n-2)\epsilon_n-n\epsilon_{n+1} \\ &=n\alpha_1+2(n-1)\alpha_2+\ldots+i(n-i+1)\alpha_i+\ldots+n\alpha_n \end{align}

Cartan matrix

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