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$\DeclareMathOperator\Pic{Pic}$Let $G$ be be a connected reductive group over a field $k$ of characteristic 0. Let $F\subseteq G$ be a connected algebraic subgroup, and set $Y=G/F$. By Proposition 6.10 in Sansuc's paper Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres there is a natural exact sequence of abelian groups $$ {\sf X}^*(G)\to {\sf X}^*(F)\to \Pic Y\to \Pic G, $$ where ${\sf X}^*(G)$ denotes the character group of $G$, and the map ${\sf X}^*(G)\to {\sf X}^*(F)$ is the restriction homomorphism. In your case $G=\operatorname{GL}(n)$, and hence $\Pic G=0$. We obtain a canonical isomorphism $$ \Pic Y\cong \operatorname{coker}\left[{\sf X}^*(G)\to {\sf X}^*(F)\right]. $$