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classification of indecomposable root systems

There are four infinite families of indecomposable root systems :

$\displaystyle A_n$	$\displaystyle =\{ \pm e_i\mp e_j \colon 1\leq i<j\leq n+1\};$
$\displaystyle B_n$	$\displaystyle =\{\pm e_i\pm e_j \colon 1\leq i<j\leq n\}\cup \{ \pm e_i \colon 1\leq i\leq n\};$
$\displaystyle C_n$	$\displaystyle =\{\pm e_i\pm e_j \colon 1\leq i<j\leq n\}\cup\{\pm 2e_i \colon 1\leq i\leq n\};$
$\displaystyle D_n$	$\displaystyle =\{\pm e_i\pm e_j \colon 1\leq i<j\leq n\}$

The subscript on the name of the root system is the dimension of $\Eset$ , the ambient Euclidean space containing the root system. In the case of $A_n$ , the ambient $\Eset$ is the $n$ -dimensional subspace perpendicular to $\sum_{i=1}^n e_i$ . In the other 3 cases, $\Eset=\Rset^n$ . Throughout, we endow $\Rset^n$ with the standard Euclidean inner product, and let $e_i,\; 1\leq i\leq n$ denote the standard basis.

As well, there are 5 exceptional, crystallographic root systems:

$\displaystyle G_2$	$\displaystyle =A_3 \cup \left\{ \pm\frac{1}{3}( 2 e_1 -e_2-e_3), \pm\frac{1}{3}( - e_1 +2e_2-e_3),\pm\frac{1}{3}( - e_1 -e_2+2e_3)\right\};$
$\displaystyle F_4$	$\displaystyle =B_4 \cup \left\{ \frac{1}{2}(\pm e_1\pm e_2\pm e_3 \pm e_4)\right\};$
$\displaystyle E_6$	$\displaystyle = A_6 \cup \left\{ \pm(e_7-e_8) \} \cup \{ \frac{1}{2}( \sum_{i=1}^6 (\pm e_i) \pm (e_7 - e_8)) \colon \text{4 minus signs}\right\};$
$\displaystyle E_7$	$\displaystyle =A_8\cup \left\{\sum_{i=1}^8 (\pm e_i) \colon \text{4 minus signs}\right\};$
$\displaystyle E_8$	$\displaystyle =D_8\cup \left\{\sum_{i=1}^8 (\pm e_i) \colon \text{even number of minus signs}\right\}.$