Geometric Representation Theory (Lecture 4)
Posted by John Baez
You can now watch lecture 4 of our seminar on Geometric Representation Theory. What happened to lectures 2 and 3, you ask? James Dolan gave those while I was travelling, but the videos aren’t available yet — sorry! Luckily, we’re tackling this subject from slightly different angles, so you can follow my latest lecture before watching his:

Lecture 4 (Oct. 9)  John Baez on categorifying and $q$deforming the theory of binomial coefficients — and multinomial coefficients! — using the analogy between projective geometry and set theory. Review of uncombed Young diagrams $D$, and $D$flags on finite sets and finitedimensional vector spaces over the field with $q$ elements, $F = F_q$. When $D$ has $n$ boxes, two rows, and just one box in the first row, the set $D(F^n)$ is the $(n1)$dimensional projective space over $F$, and the number of points in $D(F^n)$ is the $n$th $q$integer, defined by:
$[n]_q = \frac{q^n  1}{q  1}$
When $D$ has $n$ boxes, two rows, and $k$ boxes in the first row, $D(F^n)$ is the Grassmannian consisting
of $k$dimensional subspaces of $F^n$, and the number of points in $D(F^n)$ is the $q$binomial coefficient
$\binom{n}{k}_q = \frac{[n]!_q}{[k]!_q [nk]!_q}$
where the $q$factorial $[n]!_q$ is given by
$[n]!_q = [1]_q [2]_q \cdots [n]_q$
For a general uncombed Young diagram
$D$, $D(F^n)$ is a partial flag variety, and its number of points
is a ‘$q$multinomial coefficient’. Young subgroups versus parabolic
subgroups. Decomposing projective spaces into Schubert cells.

Streaming
video in QuickTime format; the URL is
http://mainstream.ucr.edu/baez_10_9_stream.mov  Downloadable video (694 megabytes)
 Lecture notes by Alex Hoffnung

Streaming
video in QuickTime format; the URL is
Handwritten Notes
Apoorva Khare and Chris Rogers are also taking handwritten notes, which should eventually show up here. We can vote on who is best.
Videos
We’re offering the videos in streaming and/or downloadable form, both as .mov files. Downloading them takes a long time, but you may need to do this, since the streaming videos seem to work well only if you have a good internet connection.
.mov files can best be played using a free program called QuickTime. If you have QuickTime and your web browser has .mov files associated to this program, you should be able to click on the “streaming video” link above and watch the video. An alternate method is to launch the QuickTime player on your computer, click on “File” and then “Open URL”, and type in the URL provided above. This has the advantage that you can easily make the picture bigger.
If you can handle URL’s that begin with rtsp, you can instead go the corresponding URL of that form, e.g. rtsp://mainstream.ucr.edu/baez_10_9_stream.mov. This may also have advantages, but at present my computer gags on such URL’s.
If you encounter problems or — even better — know cool tricks to solve such problems, please let us know about them here!
Errata
If you catch mistakes, let me know and I’ll add them to the list of errata.
Re: Geometric Representation Theory (Lecture 4)
I had to upgrade my quicktime to v7.2 in order to get the videostream from “http://mainstream.ucr.edu/baez_10_9_stream.mov”, after which “rtsp://mainstream.ucr.edu/baez_10_9_stream.mov” played fine as well. Both in firefox and IE