to prove these results and we state the main theorem which is the basis for the homological stability proofs. Consider a group $G$ with a weak spherical Tits system of rank . We enumerate the type set $I=\{i_1,\ldots,i_{n+1}\}$ arbitrarily. For $1\leq p\leq n+1$, denote by $L_p$ certain Levi subgroups of $G$ of type $I\backslash\{i_p\}$.

For the applications discussed in the previous section, the group $G$ is of type $A_{n+1}$ or $C_{n+1}$ with a linear ordering of the type set. The resulting Coxeter diagrams of $G$ and $L_p$ are illustrated in the following picture.

\(G) at (-1,.8) [$G$]{}; (L) at (-1,0) [$L_p$]{}; in [0,.8]{} [ in [0,1,2,3,4,6,7,8,9,10]{} [ (,) circle (.7mm);]{} (1,) – (2,); (2,) – (3,); (3,) – (4,); (6,) – (7,); (7,) – (8,); (8,) – (10,); (0,+ .05) – (1,+ .05); (0,- .05) – (1,-.05); ]{} (4,.8) – (6,.8); (5,.8) circle (.7mm); (0) at (0,-.6) [$1$]{}; (1) at (1,-.6) [$2$]{}; (3) at (4,-.6) [$p-1$]{}; (4) at (5,-.63) [$p$]{}; (5) at (6,-.6) [$p+1$]{}; (7) at (9,-.62) [$n$]{}; (8) at (10,-.6) [$n+1$]{};

We choose a subgroup $G'\leq L_{n+1}$ of type $I\backslash \{i_{n+1}\}$ and write $L'_p=L_p\cap G'$. Again, in the concrete applications, we have the following situation.

\(G) at (-1,.8) [$G'$]{}; 