Compositions, partial flags, and standard parabolics

Compositions. A positive-part composition of $n$, or simply a composition, is a finite list $\alpha =(a_1,\dots ,a_r)$ of positive integers with $a_1+\cdots +a_r=n$; here $r\ge 0$ is the length of $\alpha$ and the $a_i$ are its parts. A composition of $n$ thus records an ordered partition of $n$ into nonzero parts, and for $n\ge 1$ the length satisfies $1\le r\le n$. Its partial sums are $d_i:=a_1+\cdots +a_i(0\le i\le r),$ so that $0=d_0<d_1<\cdots <d_r=n$. The blocks of $\alpha$ are the disjoint intervals of indices $I_i:=\{d_{i-1}+1,d_{i-1}+2,\dots ,d_i\}\subseteq \{1,\dots ,n\}(1\le i\le r),$ which cover $\{1,\dots ,n\}$ in order and have $\mathrm{\mid }{I}_i\mathrm{\mid }=a_i$. Two extreme cases occur throughout: the one-part composition $\alpha =(n)$, with $r=1$ and block $\{1,\dots ,n\}$, and the all-singletons composition $\alpha =(1,1,\dots ,1)=(1^n)$, with $r=n$ and $I_i=\{i\}$. Write $\mathrm{blk}(k):=i$ for the unique $i$ with $k\in I_i$, so that $k<j$ with $\mathrm{blk}(k)<\mathrm{blk}(j)$ means that $k$ lies in a strictly earlier block than $j$.

Partial flags of type $\alpha$. Fix the $n$-dimensional ${\mathbb{F}}_q$-space $V={\mathbb{F}}_q^n$ with standard basis $e_1,\dots ,e_n$ and standard flag $V_0⊊V_1⊊\cdots ⊊V_n=V$, where $V_j=⟨e_1,\dots ,e_j⟩$ and ${\dim }_{{\mathbb{F}}_q}{V}_j=j$ (Standard subgroups of finite general linear groups). A partial flag of type $\alpha$ is a strictly increasing chain of linear subspaces $0=F_0⊊F_1⊊\cdots ⊊F_r=V,{\dim }_{{\mathbb{F}}_q}{F}_i=d_i,$ that is, a chain whose successive dimensions are the partial sums of $\alpha$; equivalently, a chain of subspaces of $V$ in which exactly the dimensions $d_0,\dots ,d_r$ occur. The standard partial flag of type $\alpha$ is $W_{\bullet }^{(\alpha )}:W_i:=V_{d_i}=⟨e_1,\dots ,e_{d_i}⟩,{\dim }_{{\mathbb{F}}_q}{W}_i=d_i,$ and its members are standard coordinate subspaces; the standard flag of Standard subgroups of finite general linear groups is the special case $\alpha =(1^n)$. Complete flags correspond to the composition $(1,1,\dots ,1)$ and arbitrary subspaces of dimension $r$ to $(r,n-r)$ with $1\le r<n$.

The standard parabolic subgroup of type $\alpha$. A matrix $p=(p_{kl})\in M_n({\mathbb{F}}_q)$ is upper block triangular of type $\alpha$ when $p_{kl}=0\text{whenever }\mathrm{blk}(k)>\mathrm{blk}(l),$ that is, whenever $k$ lies in a strictly later block than $l$; entries with $\mathrm{blk}(k)\le \mathrm{blk}(l)$ are unconstrained. Thus $p$ is upper block triangular of type $\alpha$ exactly when it has the block shape $\left(\begin{array}{cccc}{A}_{11}& A_{12}& \cdots & A_{1r}\\ 0& A_{22}& \cdots & A_{2r}\\ \mathrm{⋮}& \ddots & \ddots & \mathrm{⋮}\\ 0& \cdots & 0& A_{rr}\end{array}\right),A_{ij}\in M_{a_i\times a_j}({\mathbb{F}}_q),$ with square diagonal blocks $A_{ii}$ and arbitrary blocks above the diagonal. The standard parabolic subgroup of type $\alpha$ is $P_{\alpha }:=\{p\in G:p\text{ is upper block triangular of type }\alpha \},$ and it is a subgroup of $G$: it contains $I_n$, and sums and products of upper block triangular matrices, as well as inverses of invertible ones, are again upper block triangular, because the block shape is stable under the block matrix operations of Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication. It is the stabiliser in $G$ of the standard partial flag of type $\alpha$, as verified below, and the two extreme cases are $P_{(n)}=G,P_{(1^n)}=B,$ the second because "upper block triangular of type $(1^n)$" says $p_{kl}=0$ for $k>l$, which is exactly the definition of an upper triangular matrix (Upper triangular, lower triangular and diagonal square matrices over a commutative ring), whose invertible members form $B$ by Standard subgroups of finite general linear groups. The subgroup $P_{\alpha }$ is the standard parabolic subgroup containing $B$, and $B\le P_{\alpha }\le G$ for every composition $\alpha$.

Partial flags as the coset space $G\mathrm{/}{P}_{\alpha }$. The group $G$ acts on the set ${\mathcal{F}}_{\alpha }$ of partial flags of type $\alpha$ by $g\cdot F_{\bullet }:=(g(F_0),\dots ,g(F_r))$; this is a well-defined left action because $g$ is an invertible linear map, so it preserves strict inclusions and dimensions, with $I_n$ acting trivially. The action is transitive: given $F_{\bullet }\in {\mathcal{F}}_{\alpha }$, choose vectors as follows. Extending a basis of $F_{i-1}$ to a basis of $F_i$ is possible by the extension clause of If ${\dim }_{F}V=n$ and $U$ is a linear subspace of $V$, then $U$ is finite-dimensional, ${\dim }_{F}U\le n$, and ${\dim }_{F}U=n$ if and only if $U=V$, and doing this successively for $i=1,\dots ,r$ produces an ordered basis $v_1,\dots ,v_n$ of $V$ with $F_i=⟨v_1,\dots ,v_{d_i}⟩(0\le i\le r);$ the linear map $g$ with $g(e_j)=v_j$ then has an invertible matrix in the standard basis (Complete flags are G/B, where the same argument produces an element of $G$) and satisfies $g(W_i)=F_i$ for all $i$, that is, $g\cdot W_{\bullet }^{(\alpha )}=F_{\bullet }$. The stabiliser of the standard partial flag is $P_{\alpha }$: a matrix $p\in G$ satisfies $p(W_i)=W_i$ for all $i$ exactly when $p(W_i)\subseteq W_i$ for all $i$, because $p$ is injective and ${\dim }_{{\mathbb{F}}_q}p(W_i)={\dim }_{{\mathbb{F}}_q}{W}_i=d_i$; and $p(W_i)\subseteq W_i$ means that the column block $l$ of $p$ has no nonzero entries below block $i$ whenever $\mathrm{blk}(l)\le i$, which is exactly $p_{kl}=0$ for $\mathrm{blk}(k)>\mathrm{blk}(l)$. Consequently the orbit map $G\mathrm{/}{P}_{\alpha }⟶{\mathcal{F}}_{\alpha },g{P}_{\alpha }⟼g\cdot W_{\bullet }^{(\alpha )},$ is a $G$-equivariant bijection by Every transitive $G$-set is equivariantly isomorphic to $G\mathrm{/}{G}_x$ for any chosen point $x$ and Left group actions, transitive actions, and faithful actions, so partial flags of type $\alpha$ are the left cosets of $P_{\alpha }$.

Block Levis and unipotent radicals. With $\alpha$ fixed, the standard Levi subgroup of type $\alpha$ is the group of block diagonal matrices $L_{\alpha }:=\{l\in P_{\alpha }:l_{kl}=0\text{ whenever }\mathrm{blk}(k)\mathrm{\ne }\mathrm{blk}(l)\},$ isomorphic to ${\mathrm{GL}}_{a_1}({\mathbb{F}}_q)\times \cdots \times {\mathrm{GL}}_{a_r}({\mathbb{F}}_q)$ by the block diagonal blocks, and the standard unipotent radical is $U_{\alpha }:=\{u\in P_{\alpha }:u_{kk}=1\text{ for all }k,\text{ and }{u}_{kl}=0\text{ whenever }k\mathrm{\ne }l\text{ and }\mathrm{blk}(k)\ge \mathrm{blk}(l)\},$ the upper block triangular matrices with identity diagonal blocks and the off-diagonal blocks above the block diagonal. Thus $U_{(n)}=\{I_n\}$ and $U_{(1^n)}=U$ is the standard maximal unipotent subgroup of $G$. The decomposition $P_{\alpha }=L_{\alpha }⋉U_{\alpha }$ and the identification $L_{\alpha }\cong {\prod }_{i\le r}{\mathrm{GL}}_{a_i}({\mathbb{F}}_q)$ are proved in Block Levi decomposition of standard parabolics.