November 10, 2016

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.

Posted at November 10, 2016 2:13 AM UTC

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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 $P \geq B$.

Posted by: Allen Knutson on November 11, 2016 3:24 AM | Permalink | Reply to this

Re: Linear Algebraic Groups (Part 7)

Thanks! I haven’t been proving a lot of the important technical results, instead contenting myself to state big general theorems and illustrate them in the case of $\mathrm{GL}(n)$. The problem is that the routes to these big general theorems that I’ve seen (in Springer and Milne) are very lengthy, and I just don’t have time in a one-quarter course where my focus is on Klein geometry. I want to teach this course again and fill in more details. So, tips like this are very helpful.

Do you think the result you mention is the main ‘virtue’ of connected solvable affine algebraic groups?

Posted by: John Baez on November 11, 2016 3:47 PM | Permalink | Reply to this

Re: Linear Algebraic Groups (Part 7)

Is this related to the special place that Euclidean and Minkowskian geometry have within the family of general projective metrics? (degenerate ideal domain, resp. the “circular points at infinity” and the light cone)..

-drl

Posted by: DRLunsford on November 15, 2016 2:31 PM | Permalink | Reply to this

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