Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 111 / 4 / b / Solution

Write the basic construction aswhere $(w,x)\sim (w^{\prime },x)$ exactly when $w^{-1}{w}^{\prime }$ belongs to the subgroup generated by the mirrors containing $x$. Let $v_T$ be the vertex of $K$ corresponding to a spherical subset of a Coxeter system $T$. Its stabilizer is $W_T$. ThereforeEvery conjugate of every spherical standard parabolic subgroup consequently occurs as a vertex stabilizer.