TITLE: What are the length of the longest element in a Coexter group for every type?
QUESTION [6 upvotes]: What are the length of the longest element in a Coxeter group for every type? Thank you very much.

REPLY [8 votes]: 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