### Linear Algebraic Groups (Part 7)

#### Posted by John Baez

One of the less obvious but truly fundamental realizations in group theory is the importance of the ‘parabolic subgroups’ of a linear algebraic group. Today we’ll sneak up on this realization using the example of $\mathrm{GL}(n)$.

We’ve already seen the Klein geometry corresponding to this group has important kinds of figures — points, lines, planes, etc. — whose stabilizers are certain nice groups called ‘maximal parabolic subgroups’ of $\mathrm{GL}(n)$. But there are also important figures build from these, like ‘a point lying on a line’, or ‘a line lying on a plane’. These are called ‘flags’, and their stabilizers are called ‘parabolic subgroups’. Today we’ll work out what these parabolic subgroups of $\mathrm{GL}(n)$ are like. Especially important is the smallest one, called the ‘Borel’.

With this intuition in hand, we’ll want to generalize all these concepts to an *arbitrary* linear algebraic group. Amazingly, you can just hand someone such a group, and they can *figure out* the important kinds of geometrical figures in its Klein geometry, by determining its parabolic subgroups!

- Lecture 7 (Oct. 13) - Flags, and the flag variety $F(n_1, \dots, n_\ell, n)$, which consists of all chains of linear subspaces $V_1 \subset V_2 \subset \cdots \subset V_\ell \subset k^n$. The flag varieties as quotients of the general linear group by parabolic subgroups, which are intersections of maximal parabolic subgroups. The complete flag variety $F_n = F(1,2,\dots,n)$ as the quotient $GL(n)/B(n)$ where $B(n) = P(1,2,\dots,n)$ is the group of invertible upper triangular matrices, also called the Borel subgroup of $GL(n)$. The cardinality of the complete flag variety over $\mathbb{F}_q$ is the $q$-factorial $[n]_q!$. When $q = 1$ this reduces to the ordinary factorial, which counts ‘set-theoretic flags’.

As ever, the notes are due to John Simanyi. If you find mistakes, please let me know.

## Re: Linear Algebraic Groups (Part 7)

It took me some years to decide that the basic theorem to start with is Borel’s, that if a connected affine solvable group $B$ acts on a complete (nonempty) variety $X$, then there must be a fixed point. (Proof: $B$ has a subnormal chain with $1$-dimensional affine subquotients, so the problem reduces to looking at such $1$-d groups acting on smooth curves. Then those curves have to be genus $0$, and there are only two cases to check.)

Hence if $G$ non-solvable acts on $X$ complete, any minimal orbit $G/P$ must be complete, hence $B$ (maximal solvable inside $G$) must have a fixed point on $G/P$. So up to conjugacy, $P$ contains $B$. It’s not as obvious that any $G/P$ is complete, just from $P \geq B$.