### Linear Algebraic Groups (Part 3)

#### Posted by John Baez

This time we touch on some other aspects of algebraic group theory, again using the example of projective geometry. We describe the decomposition of projective space into ‘Bruhat cells’. These let us count the points of projective spaces over finite fields, which gets us a wee bit deeper into the fascinating and somewhat mysterious topic of ‘$q$-mathematics’.

As before, you can read John Simanyi’s wonderful notes in LaTeX. If you find mistakes, please let me know.

- Lecture 3 (Sept. 29) - The Schubert decomposition of $k\mathrm{P}^n$ into Bruhat cells. Examples: the real projective line $\mathbb{R}\mathrm{P}^1$, the complex projective plane $\mathbb{C}\mathrm{P}^1$ and the real projective plane $\mathbb{R}\mathrm{P}^2$. Projective geometry over finite fields: for any prime power $q$, there is a field $\mathbb{F}_q$ with $q$ elements, and the cardinality of $\mathbb{F}_q\mathrm{P}^n$ is the $q$-integer $[n]_q$.