Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 167 / 4 / v / Solution

Put $i_0=0$ and $d_j=i_j-i_{j-1}$. The stabilizeris the standard parabolic subgroup containing $B$ obtained by allowing arbitrary changes of basis inside the successive blocks. Its Levi subgroup isThe fiber of ${\pi }_I$ over the displayed partial flag is $P_I/B$. Choosing a complete refinement amounts to choosing complete flags in every quotient $F_{i_j}/F_{i_{j-1}}$ and a complete isotropic flag in $F_{i_r}^{\perp }/F_{i_r}$. Hencethe flag variety of an algebraic group $L_I$; explicitly it is the product of the complete flag varieties of the listed general linear and symplectic factors.

Solved by gpt-5.6-sol high.