How to calculate total number of roots in an A-type root system?

Question

In particular, I am interested in $A_4$ root system. Considering simpler cases of $A_2$ and $A_3$ my guess would be $(n+1)^2-(n+1)$ (where n is rank of the root system), but I'm not certain if it's true for higher ranks.

Answer 1

The Cartan decomposition of a simple Lie algebra $L$ of type $A_n$ implies that $$ |\Phi|=\dim L-\dim H=(n+1)^2-1-n=n^2+n $$ Here $\dim A_n=(n+1)^2-1$ and $\dim H$, the dimension of a Cartan subalgebra $H$, is equal to the rank of $A_n$, which is $n$. In fact, $H$ consists of diagonal matrices of size $n+1$, but with trace zero. So your result is correct.