Grassmannians as maximal parabolic quotients

The composition $\alpha =(r,n-r)$ has length $2$, partial sums $d_1=r$ and $d_2=n$, and blocks $I_1=\{1,\dots ,r\}$, $I_2=\{r+1,\dots ,n\}$; hence a partial flag of type $\alpha$ is a chain $0=F_0⊊F_1⊊F_2=V$ with ${\dim }_{{\mathbb{F}}_q}{F}_1=r$, and its last member is forced to be $V$. Consequently $F_{\bullet }\mapsto F_1$ is a bijection from ${\mathcal{F}}_{\alpha }$ onto the set $\mathrm{Gr}(r,n)$ of $r$-dimensional subspaces of $V$, whose inverse sends $E$ to the chain $0⊊E⊊V$; the standard partial flag is $W_1=V_r=⟨e_1,\dots ,e_r⟩$, $W_2=V_n=V$.

The entry criterion of [F2] for $\alpha =(r,n-r)$ reads $p_{kl}=0$ whenever $\mathrm{blk}(k)>\mathrm{blk}(l)$, that is, whenever $k\ge r+1$ and $l\le r$; hence $P_{\alpha }$ consists exactly of the matrices $p\in G$ with zero bottom-left block, that is, of the invertible block matrices $\left(\begin{array}{cc}A& B\\ 0& D\end{array}\right)$ with $A\in M_r({\mathbb{F}}_q)$, $D\in M_{n-r}({\mathbb{F}}_q)$ and arbitrary $B\in M_{r\times (n-r)}({\mathbb{F}}_q)$. In particular $P_{\alpha }$ is the stabiliser of $W_1$, because $W_2=V$ is fixed by every element of $G$ and the stabiliser of the standard partial flag is $P_{\alpha }$ by [F2].

Maximality among standard parabolics. For a composition $\beta$ of $n$ put $T(\beta ):=\{(k,l):{\mathrm{blk}}_{\beta }(k)>{\mathrm{blk}}_{\beta }(l)\}$, so that by the entry criterion of [F2] a matrix $p\in G$ lies in $P_{\beta }$ exactly when $p_{kl}=0$ for all $(k,l)\in T(\beta )$. For $k\mathrm{\ne }l$ let ${\tau }_{kl}(c):=I_n+c{e}_k{e}_l^{\mathsf{T}}$ be the elementary matrix with the single off-diagonal entry $c$ in position $(k,l)$; by [F5] one has ${\tau }_{kl}(c){\tau }_{kl}(-c)=I_n$, so ${\tau }_{kl}(c)\in G$, and ${\tau }_{kl}(c)\in P_{\beta }$ if and only if $(k,l)\mathrm{\notin }T(\beta )$ or $c=0$; taking $c=1$, the matrix ${\tau }_{kl}(1)$ lies in $P_{\beta }$ exactly when $(k,l)\mathrm{\notin }T(\beta )$.

Composing the bijection $g{P}_{\alpha }\mapsto g\cdot W_{\bullet }^{(\alpha )}$ of [F2] with the identification ${\mathcal{F}}_{\alpha }\to \mathrm{Gr}(r,n)$, $F_{\bullet }\mapsto F_1$, of step 1.1 gives the map $\phi (g{P}_{\alpha }):=g(W_1)=⟨g(e_1),\dots ,g(e_r)⟩$; it is a bijection $G\mathrm{/}{P}_{\alpha }\to \mathrm{Gr}(r,n)$ by [F2] and step 1.1, it is well defined and has singleton fibres, while the unquotiented map $g\mapsto g(W_1)$ has fibres the left cosets of $P_{\alpha }$ by step 1.2, and it is $G$-equivariant: $\phi (x\cdot g{P}_{\alpha })=\phi (xg{P}_{\alpha })=(xg)(W_1)=x(g(W_1))=x\cdot \phi (g{P}_{\alpha })$, where the action on $\mathrm{Gr}(r,n)$ is the natural one induced by the action on chains. Thus the $r$-dimensional subspaces of $V$ are the left cosets of $P_{\alpha }$ in $G$, with $g{P}_{\alpha }\leftrightarrow g(W_1)$, and $\phi$ is an isomorphism of $G$-sets.

If $(k,l)\in T(\beta )$ but $(k,l)\mathrm{\notin }T(\alpha )$, then ${\tau }_{kl}(1)$ lies in $P_{\alpha }$ but not in $P_{\beta }$ by step 1.3, so $P_{\beta }\supseteq ̸P_{\alpha }$; therefore $P_{\beta }\supseteq P_{\alpha }$ forces $T(\beta )\subseteq T(\alpha )$. Conversely $T(\beta )\subseteq T(\alpha )$ means that every matrix vanishing on $T(\alpha )$ vanishes on $T(\beta )$, that is, $P_{\alpha }\subseteq P_{\beta }$; hence $P_{\beta }\supseteq P_{\alpha }⟺T(\beta )\subseteq T(\alpha )\mathrm{.}$

The inclusion $T(\beta )\subseteq T(\alpha )$ holds if and only if every block of $\alpha$ is contained in a block of $\beta$: if $k<l$ lie in a common block of $\alpha$ then $(l,k)\mathrm{\notin }T(\alpha )$, so $(l,k)\mathrm{\notin }T(\beta )$, which by ${\mathrm{blk}}_{\beta }(l)\ge {\mathrm{blk}}_{\beta }(k)$ gives ${\mathrm{blk}}_{\beta }(l)={\mathrm{blk}}_{\beta }(k)$, so the whole $\alpha$-block lies in one $\beta$-block; conversely, if every $\alpha$-block lies in a $\beta$-block, then ${\mathrm{blk}}_{\beta }(k)>{\mathrm{blk}}_{\beta }(l)$ puts the $\beta$-block of $k$ strictly to the right of that of $l$, hence the $\alpha$-block of $k$ strictly to the right of that of $l$, that is ${\mathrm{blk}}_{\alpha }(k)>{\mathrm{blk}}_{\alpha }(l)$. For $\alpha =(r,n-r)$ the blocks are the two nonempty intervals $I_1$ and $I_2$, so a composition $\beta$ such that every $\alpha$-block lies in a $\beta$-block is either $\beta =\alpha$ or $\beta =(n)$; by step 2.2 the standard parabolics containing $P_{\alpha }$ are therefore exactly $P_{\alpha }$ and $P_{(n)}=G$, so $P_{(r,n-r)}$ is maximal among the standard parabolics $P_{\beta }$.

For $r=1$ a subspace is a line exactly when it is one-dimensional; for $0\mathrm{\ne }w\in V$ the span $⟨w⟩=\{\lambda w:\lambda \in {\mathbb{F}}_q\}$ is such a line, since $w$ is linearly independent, and every line $L=⟨v⟩$ with $0\mathrm{\ne }v$ consists of $0$ together with the $q-1$ vectors $\lambda v$ with $\lambda \mathrm{\ne }0$, which are pairwise distinct by [F6]. Every nonzero vector $w$ lies in the line $⟨w⟩$, and if $w$ lies in lines $L,L^{\mathrm{\prime }}$ then $L=⟨w⟩=L^{\mathrm{\prime }}$ by [F3], because both are one-dimensional subspaces containing the nonzero vector $w$; hence the $q^n-1$ nonzero vectors of $V$ are partitioned into the sets $L\setminus \{0\}$ of the lines, each of size $q-1$. Counting gives $q^n-1=\mathrm{\#}\mathrm{Gr}(1,n)\cdot (q-1)$, so $\mathrm{\#}\mathrm{Gr}(1,n)=(q^n-1)\mathrm{/}(q-1)=q^{n-1}+\cdots +q+1$; in particular ${\mathbb{F}}_q^2$ has $q+1$ lines and ${\mathbb{F}}_q^3$ has $q^2+q+1$ lines.

At $n\ge 2$ and $1\le r<n$ the quotient $G\mathrm{/}{P}_{(r,n-r)}$ is in $G$-equivariant bijection with the Grassmannian $\mathrm{Gr}(r,n)$ of $r$-dimensional subspaces of ${\mathbb{F}}_q^n$ via $g{P}_{\alpha }\mapsto g⟨e_1,\dots ,e_r⟩$ (step 2.1), the subgroup $P_{(r,n-r)}$ is the stabiliser of $⟨e_1,\dots ,e_r⟩$ (step 1.2) and is maximal among the standard parabolics, its only standard overgroup being $P_{(n)}=G$ (step 3.1); in the extreme case $r=1$ the quotient is the projective space of lines, of cardinality $(q^n-1)\mathrm{/}(q-1)$ (step 3.2). ∎