$\begingroup$What are the length of the longest element in a Coxeter group for every type? Thank you very much.
$\endgroup$11 Answer
$\begingroup$The length of the longest element is exactly the number of positive roots, so
- Type $A_n$: ${n + 1} \choose 2$
- Types $B_n, C_n$: $n^2$
- Type $D_n$: $n^2 - n$
- Type $I_2(n)$: n
- Type $E_6$: 36
- Type $E_7$: 63
- Type $E_8$: 120
- Type $F_4$: 24
- Type $G_2$: 6
- Type $H_3$: 15
- Type $H_4$: 60
