\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 $\mathrm{ A }_{ 4 }$

Basic information
Coxeter number
$n+1=5$
dimension of group
$n^2+2n=24$
exponents
$1,2,\ldots,n=1,2,3,4$
Weyl group

$\mathrm{S}_{ 5}$, the symmetric group on $n+1=5$ elements

permuting the $\epsilon_i$

order of the Weyl group
$(n+1)!=5!=120$
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
$n(n+1)=20$
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
Cartan matrix
\begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \\ \end{pmatrix}
determinant
$n+1=5$