The n-Category Café

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.