Standard parabolics in GL_n

The roots and root groups of ${\mathrm{G}\mathrm{L}}_n$ are as in [F1]: the weight of the coordinate function ${\chi }_i$ on $D_n$ is ${\chi }_i(\mathrm{diag}(x_1,\dots ,x_n))=x_i$; the adjoint action on $E_{ij}$ gives the weight ${\chi }_i-{\chi }_j$, and the root group $U_{ij}$ consists of the elementary matrices $I+a{E}_{ij}$, isomorphic to ${\mathbf{G}}_a$; the positive roots relative to $B$ are those with $i<j$, and the simple roots ${\alpha }_{i,i+1}$ form the base of the root system $A_{n-1}$.

Let $I\subseteq \{1,\dots ,n-1\}$ and let the complement of $I$ in the ordered set of simple roots be the partial sums listed, so that the integers $a_1,\dots ,a_s\ge 1$ sum to $n$. Choose a cocharacter with diagonal exponents constant within each block and strictly decreasing between consecutive blocks. Entrywise conjugation scales $x_{ij}$ by the exponent difference, so its limit subgroup is precisely the group with zero blocks below the diagonal and its limit kernel has identity diagonal blocks. By [F2] its zero simple-root pairings recover $I$. Thus $P_I$ is the group of invertible matrices preserving the flag of subspaces $0\subseteq k^{a_1}\subseteq k^{a_1+a_2}\subseteq \cdots \subseteq k^n$, i.e. the block upper triangular matrices with diagonal blocks of sizes $a_1,\dots ,a_s$; its unipotent radical $R_u(P_I)$ has identity diagonal blocks, zeros below and arbitrary blocks above and its Levi factor is the block diagonal ${\prod }_i{\mathrm{G}\mathrm{L}}_{a_i}$; the multiplication map $R_u(P_I)⋊L_I\to P_I$ is a group-scheme isomorphism by [F2], and $P_{\mathrm{\varnothing }}=B$ and $P_{\mathrm{\Delta }}=G$ because the corresponding flags are the complete flag and the trivial flag. The all-algebra matrix equations verify these stabilizers scheme-theoretically. For $n=1$ the simple-root set is empty, both endpoints are $G=B={\mathbf{G}}_m$, and its unipotent radical is trivial.

The maximal proper parabolics correspond to $I=\mathrm{\Delta }\setminus \{i\}$, i.e. to the stabilizers of a single subspace of dimension $i$: taking $a_1=i$, $a_2=n-i$ gives the block upper triangular group with two diagonal blocks, which is the stabilizer of $k^i\subseteq k^n$, and the general $P_I$ is the intersection of these stabilizers over the cuts $i\in \mathrm{\Delta }\setminus I$, not over $I$. For $n\ge 3$, the minimal proper parabolics strictly above $B$ are $P_{\{i\}}$, since singletons are the minimal nonempty proper subsets of the base. For $n\le 2$ there is no proper parabolic strictly between $B$ and $G$; in particular, for $n=2$ the singleton gives $G$, rather than a proper parabolic. Finally, the simple-root reflections act on the characters ${\chi }_i$ by adjacent transpositions. Since they generate the Weyl group, its faithful lattice action is precisely $S_n$. The Bruhat cells of $G\mathrm{/}B$ are indexed by $S_n$, and the length $n(w)$ equals the dimension of the cell by the length formula of [F3]. For the roots ${\chi }_i-{\chi }_j$ with $i<j$, the count $\mathrm{\mid }{\mathrm{\Phi }}^{+}\cap w{\mathrm{\Phi }}^{-}\mathrm{\mid }$ is the inversion number of $w^{-1}$. This equals the inversion number of $w$: $(i,j)\mapsto (w(j),w(i))$ bijects their inversion sets. Thus the count is $\mathrm{\#}\{(i,j):i<j,w(i)>w(j)\}$ over every field.