Bruhat Decomposition and Flags over Finite Fields — Examples

At $n=1$ the three defining conditions of [F1] read: a $1\times 1$ matrix is upper triangular and diagonal always, and it is upper unitriangular exactly when its entry is $1$; hence $B=T=G$ and $U=\{I_1\}$. By [F2] the monomial matrices are all of $G$, so $N=G$, $W=N\mathrm{/}T$ is the trivial group, and it is $S_1=\{1\}$ under the isomorphism $\sigma \mapsto P_{\sigma }T$.

For the composition $\alpha =(1)$ of $n=1$ the block is $I_1=\{1\}$, so ${\mathrm{blk}}_{\alpha }(1)={\mathrm{blk}}_{\alpha }(1)$ and the criteria of [F2] impose no vanishing condition: $P_{\alpha }=L_{\alpha }=G$; the condition for $U_{\alpha }$ is $g_{11}=1$, so $U_{\alpha }=\{I_1\}$. In particular $P_{\alpha }=G=L_{\alpha }$ and $U_{\alpha }=\{I_1\}$, and the standard parabolic is the whole group.

The set $\{1\}$ has exactly one ordered partition, namely ${\gamma }_0=(\{1\})$, whose single block is $\{1\}$; a proper refinement of ${\gamma }_0$ would be an ordered partition of $\{1\}$ whose set partition differs from $\{\{1\}\}$, and no such partition exists, since an ordered partition of $\{1\}$ consists of nonempty disjoint blocks covering $\{1\}$. Hence ${\gamma }_0$ has no proper refinement at all.

The vectors of $V={\mathbb{F}}_q^1$ are the scalar multiples of $e_1$, so the only subspace different from $0$ is $V$ itself and the chain $0<V$ is the only complete flag of $V$; by [F3] the coset space $G\mathrm{/}B$ is a single point, and by $B=T=G$ of step 1.1 the standard flag is fixed by every element of $G$.

The inflation ${\mathrm{Inf}}_{L_{\alpha }}^{P_{\alpha }}$ of [F5] is taken along the identity homomorphism $P_{\alpha }\to L_{\alpha }$, since $P_{\alpha }=L_{\alpha }=G$, so it is the identity functor on complex $G$-modules; likewise ${\mathrm{Ind}}_{P_{\alpha }}^G$ is the identity functor, because $P_{\alpha }=G$. Hence $R_G^G(W)=W$ for every complex $G$-module $W$, and ${}^{\ast }\text{\hspace{0.17em}⁣}{R}_G^G(X)=X^{U_{\alpha }}=X^{\{I_1\}}=X$ for every complex $G$-module $X$: both Harish-Chandra functors attached to the one-part composition of $n=1$ are the identity functors.

By [F4] the group is the disjoint union of the cells $B{P}_{\sigma }B$ over $\sigma \in S_1=\{1\}$, so $G=B{P}_{\mathrm{i}\mathrm{d}}B=B$ by step 1.1: there is exactly one Bruhat cell, namely the cell of the identity, and it is the single right coset of $B=G$ in $G$; this matches $\mathrm{\mid }B{P}_{\mathrm{i}\mathrm{d}}B\mathrm{/}B\mathrm{\mid }=q^{\mathrm{\ell }(\mathrm{i}\mathrm{d})}=q^0=1$ of [F4] and the single point $G\mathrm{/}B$ of step 2.1.

Let $M$ be a complex $G$-module. By step 1.3 the only ordered partition of $\{1\}$ is ${\gamma }_0$, which has no proper refinement, so the vanishing condition of [F6] is vacuous and $M$ is cuspidal. If moreover $M=\chi$ is simple, then $(G,\chi )$ is a cuspidal pair, and its Harish-Chandra series consists of the simple quotients of $R_G^G(\chi )=\chi$ (step 2.2), that is of $\chi$ itself: the series is the singleton $\{\chi \}$. Every simple complex $G$-module is one-dimensional, since $G={\mathbb{F}}_q^{\times }$ is abelian, and each of them forms its own Harish-Chandra series.

Consequently, at $n=1$ the general theory takes the following form: $B=T=G$, $U=\{I_1\}$ and $W=S_1$; the complete flag variety $G\mathrm{/}B$ and the set of Bruhat cells and double cosets $B\mathrm{\backslash }G\mathrm{/}B$ each have exactly one element; the one-part composition $(1)$ has $P_{\alpha }=L_{\alpha }=G$ with trivial unipotent radical, so both Harish-Chandra functors are the identity; and every simple module is cuspidal with a one-element series. ∎

A complete flag of the two-dimensional space $V$ is a chain $0<F_1<V$ with $\dim F_1=1$, so it is determined by its member $F_1$, which is a line; conversely every line $L$ gives the complete flag $0<L<V$. Hence the complete flags correspond bijectively to the lines in $V$, and by [F3] the coset space $G\mathrm{/}B$ is in $G$-equivariant bijection with the set of lines, that is with ${\mathbb{P}}^1({\mathbb{F}}_q)$. In particular the standard flag $V_{\bullet }$ corresponds to the standard line $⟨e_1⟩$, whose stabiliser in $G$ is $B$.

Every nonzero vector of $V$ is of the form $c_1{e}_1+c_2{e}_2$ with $(c_1,c_2)\mathrm{\ne }(0,0)$, and the line it spans is $⟨e_1⟩$ when $c_2=0$ and $⟨e_2+a{e}_1⟩$ with $a=c_1{c}_2^{-1}\in {\mathbb{F}}_q$ when $c_2\mathrm{\ne }0$; the $q+1$ lines $⟨e_1⟩$ and $⟨e_2+a{e}_1⟩$, $a\in {\mathbb{F}}_q$, are pairwise distinct, because $e_2+a{e}_1$ and $e_2+a^{\mathrm{\prime }}{e}_1$ are proportional only when $a=a^{\mathrm{\prime }}$, and none of them lies in $⟨e_1⟩$. Hence ${\mathbb{P}}^1({\mathbb{F}}_q)$ has exactly $q+1$ points.

The standard line is fixed by $B$, because an invertible upper triangular matrix sends $e_1$ to $\alpha e_1$ with $\alpha \mathrm{\ne }0$; hence $\{⟨e_1⟩\}$ is a $B$-orbit, of size $1$. For $g=\left(\begin{array}{cc}\alpha & \beta \\ 0& \delta \end{array}\right)\in B$ and $a\in {\mathbb{F}}_q$ one has $g(e_2+a{e}_1)=(\beta +\alpha a)e_1+\delta e_2=\delta (e_2+\frac{\alpha a+\beta }{\delta }{e}_1)$, so $g$ sends the line $⟨e_2+a{e}_1⟩$ to the line $⟨e_2+a^{\mathrm{\prime }}{e}_1⟩$ with $a^{\mathrm{\prime }}=\frac{\alpha a+\beta }{\delta }$; given $a,a^{\mathrm{\prime }}\in {\mathbb{F}}_q$ the choices $\alpha =\delta =1$, $\beta =a^{\mathrm{\prime }}-a$ produce such an element of $B$, so $B$ is transitive on the $q$ lines of the complement. Hence the complement of the standard line is a single $B$-orbit of size $q$, and the two $B$-orbits on ${\mathbb{P}}^1({\mathbb{F}}_q)$ have sizes $1$ and $q$.

The $B$-orbits on $G\mathrm{/}B$ are the sets $B\cdot (gB)=\{bgB:b\in B\}$ of left cosets, that is exactly the quotients $BgB\mathrm{/}B$ of the double cosets $BgB$ in $G$. By [F4] the double cosets $BgB$ are exactly $B{P}_{\mathrm{i}\mathrm{d}}B=B$ and $B{P}_{s}B$; so the two $B$-orbits of step 2.1 are the quotients $B\mathrm{/}B$ and $B{P}_{s}B\mathrm{/}B$, the orbit $\{⟨e_1⟩\}$ corresponding to the identity and the complement $\{⟨e_2+a{e}_1⟩:a\in {\mathbb{F}}_q\}$ to $s$.

By [F4] the numbers of left cosets of $B$ in the two cells are $\mathrm{\mid }B{P}_{\mathrm{i}\mathrm{d}}B\mathrm{/}B\mathrm{\mid }=q^{\mathrm{\ell }(\mathrm{i}\mathrm{d})}=q^0=1$ and $\mathrm{\mid }B{P}_{s}B\mathrm{/}B\mathrm{\mid }=q^{\mathrm{\ell }(s)}=q^1=q$ by [F2]; these agree with the orbit sizes $1$ and $q$ computed in step 2.1, and their sum $1+q$ is the number of points of ${\mathbb{P}}^1({\mathbb{F}}_q)$ found in step 1.2. Thus $G\mathrm{/}B$ is the projective line with $q+1$ points, its two $B$-orbits are the standard line of size $1$ and its complement of size $q$, and they are indexed by $\mathrm{i}\mathrm{d}$ and $s$. ∎

For the three permutations $123,132,213$ with values $\sigma (1),\sigma (2),\sigma (3)$ equal to $1,2,3$ and $1,3,2$ and $2,1,3$ the defining count of [F1] gives: for $\sigma =123$ one has $r_{1,j}=j$ and $r_{i,j}=\max (0,j-i+1)$, so the rows are $[1,2,3],[0,1,2],[0,0,1]$; for $\sigma =132$ the rows are $[1,2,3]$, then $\mathrm{\#}\{k\le j:\sigma (k)\ge 2\}=[0,1,2]$, then $\mathrm{\#}\{k\le j:\sigma (k)\ge 3\}=[0,1,1]$; for $\sigma =213$ the rows are $[1,2,3]$, then $\mathrm{\#}\{k\le j:\sigma (k)\ge 2\}=[1,1,2]$ (for $j=1$ the value $\sigma (1)=2$ contributes, for $j=2$ only $k=1$ does, for $j=3$ the values $2$ and $3$ do), then $\mathrm{\#}\{k\le j:\sigma (k)\ge 3\}=[0,0,1]$.

For the three permutations $231,312,321$ with values $2,3,1$ and $3,1,2$ and $3,2,1$ the same count gives: for $\sigma =231$ the rows are $[1,2,3]$, then $\mathrm{\#}\{k\le j:\sigma (k)\ge 2\}=[1,2,2]$, then $\mathrm{\#}\{k\le j:\sigma (k)\ge 3\}=[0,1,1]$; for $\sigma =312$ the rows are $[1,2,3]$, then $[1,1,2]$, then $[1,1,1]$; for $\sigma =321$ the rows are $[1,2,3]$, then $[1,2,2]$, then $[1,1,1]$.

For each of the six permutations the intersection-dimension matrix is obtained from the rank matrix by the formula of [F2], $D_{i,j}:=j-r_{i+1,j}(w)$ with $r_{4,j}(w)=0$: using the second and third rows listed in steps 1.1 and 1.2 this gives $D_{1,j}=[1,1,1]$ for $123$ and $132$ (second rows $[0,1,2]$), $[0,1,1]$ for $213$ and $312$ (second rows $[1,1,2]$) and $[0,0,1]$ for $231$ and $321$ (second rows $[1,2,2]$); $D_{2,j}=[1,2,2]$ for $123$ and $213$ (third rows $[0,0,1]$), $[1,1,2]$ for $132$ and $231$ (third rows $[0,1,1]$) and $[0,1,2]$ for $312$ and $321$ (third rows $[1,1,1]$); and $D_{3,j}=[1,2,3]$ for all six, because $r_{4,j}=0$. Explicitly, in the order $123,132,213,231,312,321$ these are the six displayed matrices of the Example section, and the formula $r_{i+1,j}(w)=j-D_{i,j}$ recovers the rank matrix from the intersection-dimension matrix.

The six rank matrices are pairwise distinct: those of $123$ and $132$ differ in position $(3,2)$, where they are $0$ and $1$; each of those of $123,132$ differs from that of $213$ in position $(2,1)$, where $123$ and $132$ have $0$ and $213$ has $1$; $213$ and $312$ differ in position $(3,1)$, where they are $0$ and $1$; $231$ and $321$ differ in position $(3,1)$, where they are $0$ and $1$; and each of $231,321$ differs from each of $213,312$ in position $(2,2)$, where $231,321$ have $2$ and $213,312$ have $1$. Since the rank matrices of the six permutations are pairwise distinct and a rank matrix determines its double coset by [F1], the six elements of $S_3$ realise six distinct double cosets in $B\mathrm{\backslash }G\mathrm{/}B$, in agreement with the bijection of [F3].

The six intersection-dimension matrices displayed in the Example section are pairwise distinct as well: the entry $(1,1)$ equals $1$ for $123$ and $132$ and $0$ for $213,231,312,321$, so it separates these two groups; within the first group the entry $(2,2)$ is $2$ for $123$ and $1$ for $132$; and within the second group the pair of entries $((1,2),(2,1))$ takes the four distinct values $(1,1),(1,0),(0,1),(0,0)$ for $213,312,231,321$ respectively. Consequently the six permutations of $S_3$ give the six pairwise distinct relative positions of pairs of complete flags of ${\mathbb{F}}_q^3$, and the intersection dimensions ${\dim }_{{\mathbb{F}}_q}(V_i\cap w{V}_j)$ are the complete invariant of the diagonal orbit of the pair $(V_{\bullet },w{V}_{\bullet })$ by [F3]. ∎