### Linear Algebraic Groups (Part 5)

#### Posted by John Baez

Now let’s look at projective geometry from a Kleinian viewpoint. We’ll take the most obvious types of figures — points, lines, planes, and so on — and see which subgroups of $\mathrm{GL}(n)$ they correspond to. This leads us to the concept of ‘maximal parabolic subgroup’, which we’ll later generalize to other linear algebraic groups.

We’ll also get ready to count points in Grassmannians over finite fields. For that, we need the $q$-deformed version of binomial coefficients.

- Lecture 5 (Oct. 6) - Projective geometry from a Kleinian perspective. The Grassmannians $\mathrm{Gr}(n,j)$ as spaces of points, lines, planes, etc. in projective geometry. The Grassmannians as quotients of the general linear group by the maximal parabolic subgroups $P_{n,j}$. Claim: the cardinality of $\mathrm{Gr}(n,j)$ over the finite field $\mathbb{F}_q$ is the $q$-binomial coefficient $\binom{n}{j}_q$. The mysterious sense in which set theory is linear algebra over the ‘field with one element’.