Dimension of projective homogenous G-varieties For a parabolic subgroup of $P$ an algebraic group $G$, symbolized by a set of dots in the Dynkin diagramm of $G$, one obtains a quotient variety $X = G/P$.
Given $G,P$ how can one calculate the dimension of $X$ ?
Well one has to look at the number of positve roots of both groups and just subtract. Example: $B_3$ in $F_4$ gives $24-9 = 15$. So far so good.
(See  Humphreys book on page 66.)
But if i choose $A_3$ and $A_1\times A_1$ (the outer two dots) for example what do i have to do now? Subtract two times the number of positive roots of $A_1$ or multiply that number by itself?
 A: Really what you're doing in computing this dimension is subtracting the length of the longest element $w_0(P)$ of the corresponding parabolic subgroup of the Weyl group from the length of the longest element $w_0$ of the larger Weyl group. This is because there is a factorization
$$w_0=w_0^Pw_0(P)$$
where $w_0^P$ is the longest minimal-length coset representative and $\ell(w_0^P)+\ell(w_0(P))=\ell(w_0)$. $\ell(w_0^P)$ is the dimension of the variety, and the number of roots not in the parabolic subsystem.
For $A_3$ the length is $6$ (which is the number of positive roots). For $A_1\times A_1$ the length is $2$ (this can be seen by the fact that the simple roots $\alpha_1,\alpha_3$ satisfy $(\alpha_1|\alpha_3)=0$, or by the fact that the longest element of $A_1\times A_1$ is $s_1s_3$). Thus the longest minimal-length coset representative of the quotient has length $6-2=4$.
Though I don't like giving out rote rules to memorize, in general the number of roots in the root system $R_1\times R_2$ is the number of roots in $R_1$ plus the number of roots in $R_2$.
