Ordered partitions and coordinate parabolics

Ordered partitions. An ordered partition of $\{1,\dots ,n\}$ is a finite list $\gamma =(S_1,\dots ,S_r)$ of nonempty pairwise disjoint subsets $S_1,\dots ,S_r\subseteq \{1,\dots ,n\}$ with $S_1\cup \cdots \cup S_r=\{1,\dots ,n\}$. Its members are the blocks of $\gamma$, its length is $r$ (so that $1\le r\le n$ for $n\ge 1$), and for $1\le k\le n$ we write ${\mathrm{blk}}_{\gamma }(k)$ for the unique index with $k\in S_{{\mathrm{blk}}_{\gamma }(k)}$, so that $k$ lies in an earlier block than $l$ exactly when ${\mathrm{blk}}_{\gamma }(k)<{\mathrm{blk}}_{\gamma }(l)$. A composition $\alpha =(a_1,\dots ,a_r)$ of $n$ is the same datum as the ordered partition $(I_1,\dots ,I_r)$ whose blocks are its consecutive intervals $I_i=\{d_{i-1}+1,\dots ,d_i\}$ (Compositions, partial flags, and standard parabolics); every ordered partition is of this restricted form after relabelling, and the two notions are compared below. The extreme ordered partitions are the one-block partition $(\{1,\dots ,n\})$ and the all-singleton partition $(\{1\},\{2\},\dots ,\{n\})$, which correspond to the compositions $(n)$ and $(1^n)$.

Coordinate parabolics. Let $n\ge 1$, let $q$ be a prime power, put $G={\mathrm{GL}}_n({\mathbb{F}}_q)$ with diagonal torus $T$ and standard basis $e_1,\dots ,e_n$ (Standard subgroups of finite general linear groups), and let $\gamma =(S_1,\dots ,S_r)$ be an ordered partition of $\{1,\dots ,n\}$. Define $P_{\gamma }:=\{g\in G:g_{kl}=0\text{ whenever }{\mathrm{blk}}_{\gamma }(k)>{\mathrm{blk}}_{\gamma }(l)\},$ $L_{\gamma }:=\{g\in P_{\gamma }:g_{kl}=0\text{ whenever }{\mathrm{blk}}_{\gamma }(k)\mathrm{\ne }{\mathrm{blk}}_{\gamma }(l)\},$ $U_{\gamma }:=\{g\in P_{\gamma }:g_{kk}=1\text{ for all }k,\text{ and }{g}_{kl}=0\text{ whenever }k\mathrm{\ne }l\text{ and }{\mathrm{blk}}_{\gamma }(k)\ge {\mathrm{blk}}_{\gamma }(l)\},$ the coordinate parabolic, its coordinate Levi subgroup and its coordinate unipotent radical of type $\gamma$. For a composition $\alpha$ these are the standard subgroups $P_{\alpha },L_{\alpha },U_{\alpha }$ of Compositions, partial flags, and standard parabolics. We also call $F_i^{(\gamma )}:=⟨e_j:j\in S_1\cup \cdots \cup S_i⟩(0\le i\le r)$ the coordinate flag of type $\gamma$, so that $F_{\bullet }^{(\gamma )}$ is a strictly increasing chain of subspaces with ${\dim }_{{\mathbb{F}}_q}{F}_i^{(\gamma )}=\mathrm{\mid }{S}_1\mathrm{\mid }+\cdots +\mathrm{\mid }{S}_i\mathrm{\mid }$, and $P_{\gamma }$ is the stabiliser in $G$ of $F_{\bullet }^{(\gamma )}$: a matrix $p\in G$ satisfies $p[F_i^{(\gamma )}]\subseteq F_i^{(\gamma )}$ for all $i$ exactly when, for every $l$ with ${\mathrm{blk}}_{\gamma }(l)\le i$, the $l$-th column of $p$ has no nonzero entry in a block $S_{i^{\mathrm{\prime }}}$ with $i^{\mathrm{\prime }}>i$, which is exactly $p_{kl}=0$ whenever ${\mathrm{blk}}_{\gamma }(k)>{\mathrm{blk}}_{\gamma }(l)$ (the argument is that of Compositions, partial flags, and standard parabolics, where the same computation is carried out for consecutive blocks, and it uses only the block shape); since $p$ is injective and the chain is finite with ${\dim }_{{\mathbb{F}}_q}p[F_i^{(\gamma )}]={\dim }_{{\mathbb{F}}_q}{F}_i^{(\gamma )}$ (Invertible matrix theorem: invertibility, full pivot rank, RREF $I$, trivial nullspace and unique solvability are equivalent, Linear subspace of a vector space), the inclusions are equalities. In particular $P_{\gamma }$ is a subgroup of $G$, since the stabiliser of a subset of the flag is a subgroup (Subgroup, Left group actions, transitive actions, and faithful actions), and the diagonal torus satisfies $T\le L_{\gamma }\le P_{\gamma }$.

Change of coordinates. Let $\sigma \in S_n$ and let $u:=P_{\sigma }\in G$ be the associated permutation matrix (Permutation Weyl group and inversion length), so that $(u^{-1}gu{)}_{ij}=g_{\sigma (i),\sigma (j)}$ for every matrix $g$ (Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication, Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes). For an ordered partition $\gamma =(S_1,\dots ,S_r)$ put $\sigma (\gamma ):=(\sigma (S_1),\dots ,\sigma (S_r))$, again an ordered partition of $\{1,\dots ,n\}$; then ${\mathrm{blk}}_{\sigma (\gamma )}(k)={\mathrm{blk}}_{\gamma }({\sigma }^{-1}(k))(1\le k\le n),$ and applying the entry formula for conjugation to the three defining conditions above gives $P_{\sigma (\gamma )}=u{P}_{\gamma }{u}^{-1},L_{\sigma (\gamma )}=u{L}_{\gamma }{u}^{-1},U_{\sigma (\gamma )}=u{U}_{\gamma }{u}^{-1}\mathrm{.}$ For instance $g\in u{U}_{\gamma }{u}^{-1}$ says that $(u^{-1}gu{)}_{kk}=1$ and $(u^{-1}gu{)}_{kl}=0$ whenever $k\mathrm{\ne }l$ and ${\mathrm{blk}}_{\gamma }(k)\ge {\mathrm{blk}}_{\gamma }(l)$, which by the displayed formula for ${\mathrm{blk}}_{\sigma (\gamma )}$ is the defining condition for $U_{\sigma (\gamma )}$; the other two computations are identical. Since a composition $\alpha$ is an ordered partition with consecutive blocks, and since an arbitrary ordered partition $\gamma$ becomes a composition after the relabelling that replaces every block by its position, the change of coordinates shows that the coordinate parabolics of the ordered partitions are exactly the conjugates $g{P}_{\alpha }{g}^{-1}$ of the standard parabolics by the permutation matrices, and in particular they are split parabolic subgroups in the sense of Harish-Chandra induction and restriction for finite general linear groups.

Consequences of the change of coordinates. Fix an ordered partition $\gamma =(S_1,\dots ,S_r)$, choose the composition $\alpha :=(\mathrm{\mid }{S}_1\mathrm{\mid },\dots ,\mathrm{\mid }{S}_r\mathrm{\mid })$ of $n$, and let $\sigma \in S_n$ be the unique permutation with $\sigma (I_i)=S_i$ for all $i$ that is increasing on each block $I_i$; then $\gamma =\sigma (\alpha )$, and because $P_{\alpha }=L_{\alpha }⋉U_{\alpha }$ with $U_{\alpha }⊴P_{\alpha }$ (Block Levi decomposition of standard parabolics), conjugation by the permutation matrix $u=P_{\sigma }$ maps this decomposition to $P_{\gamma }=L_{\gamma }⋉U_{\gamma },U_{\gamma }⊴P_{\gamma },L_{\gamma }\cap U_{\gamma }=\{I_n\},$ because conjugation is an automorphism of $G$ and carries the block diagonal matrices of type $\alpha$ onto those of type $\gamma$ (${\mathrm{GL}}_n(F)$ is a group under matrix multiplication, including the trivial group ${\mathrm{GL}}_0(F)$, Normal subgroup: invariance under conjugation); moreover $P_{\gamma }$ is the stabiliser of the coordinate flag $F_{\bullet }^{(\gamma )}$, as computed above. For the Weyl group, a permutation $\rho \in S_n$ satisfies $\rho (S_i)=S_i$ for all $i$ exactly when $({\sigma }^{-1}\rho \sigma )(I_i)=I_i$ for all $i$, so with $W_{\gamma }:=\{\rho \in S_n:\rho (S_i)=S_i\text{ for every }1\le i\le r\}$ one has $W_{\gamma }=\sigma W_{\alpha }{\sigma }^{-1}$ in the group $S_n$, where $W_{\alpha }$ is the parabolic Weyl subgroup of Permutation Weyl group and inversion length; combining that criterion with the change of coordinates shows that a monomial matrix lies in $P_{\gamma }$ exactly when its permutation lies in $W_{\gamma }$, so that $N\cap P_{\gamma }=T\cdot \{P_{\rho }:\rho \in W_{\gamma }\}$ for the monomial subgroup $N$ of Permutation Weyl group and inversion length. Finally, varying only the order of the blocks leaves the Levi unchanged: if ${\gamma }^{\mathrm{\prime }}$ is obtained from $\gamma$ by permuting the list $(S_1,\dots ,S_r)$, then ${\mathrm{blk}}_{{\gamma }^{\mathrm{\prime }}}(k)\mathrm{\ne }{\mathrm{blk}}_{{\gamma }^{\mathrm{\prime }}}(l)$ holds exactly when ${\mathrm{blk}}_{\gamma }(k)\mathrm{\ne }{\mathrm{blk}}_{\gamma }(l)$, namely when $k,l$ lie in different blocks, so $L_{{\gamma }^{\mathrm{\prime }}}=L_{\gamma }$; the unipotent radical $U_{\gamma }$ on the other hand depends on the order of the blocks, as the case $r=2$ shows.

• Coordinate parabolics contain the diagonal torus, and for $q>2$ they are all the parabolics that do. We have $T\le L_{\gamma }\le P_{\gamma }$ from the definitions above. Conversely let $q>2$ and let $P\le G$ be a parabolic subgroup of $G$ containing $T$, say $P=g{P}_{\alpha }{g}^{-1}$ for some composition $\alpha$ and some $g\in G$; since $P_{\alpha }$ stabilises the standard partial flag $W_{\bullet }^{(\alpha )}$ of type $\alpha$ (Compositions, partial flags, and standard parabolics), the group $P$ is the stabiliser of the partial flag $\mathcal{F}:=g{W}_{\bullet }^{(\alpha )}$, and $T\le P$ means that every member of $\mathcal{F}$ is $T$-stable. Since $q>2$, choose $\lambda \in {\mathbb{F}}_q^{\times }$ with $\lambda \mathrm{\ne }1$. For each $i$ let $t_i\in T$ act by $\lambda$ on $e_i$ and by $1$ on every other coordinate line. If $W\subseteq V$ is $T$-stable and $v={\sum }_j{v}_j{e}_j\in W$, then $(t_i-1)v=(\lambda -1)v_i{e}_i\in W$, hence $v_i{e}_i\in W$ because $\lambda -1$ is invertible. Thus $W$ is spanned by the coordinate vectors it contains; therefore every member of $\mathcal{F}$ is a coordinate subspace, and the resulting strictly increasing chain of coordinate subspaces is the coordinate flag of a unique ordered partition $\gamma$ of $\{1,\dots ,n\}$ with $P=P_{\gamma }$. At $q=2$ the conclusion fails, and not merely for want of proof: the torus $T$ is then the trivial group, so every parabolic contains it, while for $n=2$ the three Borels of ${\mathrm{GL}}_2({\mathbb{F}}_2)\cong S_3$ are the upper triangular, the lower triangular and the stabiliser of the line $⟨e_1+e_2⟩$; only the upper and lower triangular ones are coordinate parabolics, those of $(\{1\},\{2\})$ and $(\{2\},\{1\})$. Consequently the theory below is stated for coordinate parabolics, which is the class that the finite general linear group controls at every $q$; the larger class of all parabolics containing $T$ agrees with it whenever $q>2$.
• Every block order is a choice. Two ordered partitions with the same blocks in different orders give the same Levi and the same $W_{\gamma }$ but different parabolics $P_{\gamma }$; comparing the corresponding Harish-Chandra functors is the content of the parabolic-independence theorem of this page, and it is the reason the parabolics, not only the Levis, are carried as data in Harish-Chandra induction and restriction for finite general linear groups.