## October 12, 2007

### 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 finite-dimensional 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 $(n-1)$-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 $\binom{n}{k}_q = \frac{[n]!_q}{[k]!_q [n-k]!_q}$ 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.

#### 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.

Posted at October 12, 2007 9:33 PM UTC

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### 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

Posted by: ericv on October 13, 2007 12:25 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 4)

Eric wrote:

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.

Interesting. Can you detect any advantages to the latter? The former works fine for me, but not the latter, so I’m not sure what I’m missing.

I had QuickTime v7.1. I upgraded to v7.2, but my Firefox still says it doesn’t know what to do with rtsp (= real-time streaming protocol).

Even the following advice which I found online didn’t help — but I’ll pass it on anyway:

What if I’m having problems viewing streaming media?

If you are using Internet Explorer or Safari, and QuickTime is installed, you should be able to play streaming media. However, if you are behind a strict firewall, you may not be able to tune in. Streaming media requires that TCP traffic on port 554, as well as UDP traffic on ports 6970 through 6999 be allowed. If you are unsure how to modify these settings, please contact your system administrator or internet service provider.

Firefox Users Note: Mozilla Firefox does not have the RTSP protocol defined by default. Typically, QuickTime will set this definition for you when it installs, however occasionally this may not be the case. In order to enable streaming media, type about:config into your browser address bar. Search for the preference network.protocol-handler.app.rtsp. If it does not exist, you will need to follow the following steps to enable it:

Step 1) If you do not have the Firefox preference listing open, type about:config into your browser address bar and hit enter.

Step 2) Right click the listing of preferences, and select New -> String.

Step 3) When prompted, enter the preference name: network.protocol-handler.app.rtsp and click “OK”.

Step 4) You will then be prompted for the string value. Enter the path to your QuickTime Player executable (This is usually C:/Program Files/QuickTime/QuickTimePlayer.exe) and click “OK”.

Step 5) Close and restart your browser.

You should now be able to enjoy streaming media with Firefox.

Windows Users Note: You must associate QuickTime with RTSP streaming media in order to view streaming media. This is usually defined by default when QuickTime has installed, but occasionally this is not the case. In order to do so, install QuickTime, and complete the following steps:

Step 1) Open the Windows Control Panel, and double-click on the QuickTime option.

Step 2) Click the “File Types” tab, and make sure the “Streaming - Streaming movies” box is checked. If it has a green box or gray arrow selection, it is probably not associated with RTSP streaming.

Step 3) Click “OK” and restart your browser.

Posted by: John Baez on October 14, 2007 1:13 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 4)

Hi,

None whatsoever really. The media streams the same way as in the http version except that it starts a new window.

Funny thing with that rtsp protocol is that it appears to need both Realplayer and Quicktime. If you are having that problem both in firefox and IE then it could be that Realplayer isn’t installed or isn’t working properly and needs to be reinstalled.

Posted by: ericv on October 14, 2007 10:15 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 4)

i wrote : “.. Funny thing with that rtsp protocol is that it appears to need both Realplayer and Quicktime”..”

What i meant was that is appears like that on my pc. it opens a realplayer window which in turn opens quicktime to play the .mov, which is kind of odd.
It should obviously work with either Real player or QT.

Posted by: ericv on October 14, 2007 10:26 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 4)

Eric wrote:

What i meant was that is appears like that on my pc it opens a realplayer window which in turn opens quicktime to play the .mov, which is kind of odd.

Yeah, that’s pretty weird, especially given the following. When I first tried accessing this files, I had Realplayer installed, and it produced a weird error message. The guy at the Multimedia Center here at UCR told me to uninstall Realplayer, so QuickTime would take over. So I did — since I hate Realplayer anyway.

This made the streaming videos work nicely when I use the http:// URLs. But, the rtsp:// URLs still don’t work.

Since you say there’s no serious difference between the two anyway, I’ll forget about rtsp for now.

Posted by: John Baez on October 15, 2007 2:54 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 4)

The streaming version came down fine for me (with the occasional bit of very strange-looking fuzz). The download version came down in about an hour, and seems fine.

Posted by: Tim Silverman on October 13, 2007 9:21 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 4)

For some reason I have problems with the streaming version of this video, which stalls after the first few seconds - I didn’t encounter that with the previous one. In any case, I’m sure all these technical things will get ironed out eventually. Keep ‘em coming!

Posted by: Jeffrey Morton on October 14, 2007 12:21 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 4)

Hmm — I don’t experience that problem.

Posted by: John Baez on October 14, 2007 1:07 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 4)

I’d really like to get comments on the math, too. Though we’re just getting started in this seminar, I already think this is wonderfully fun stuff.

Posted by: John Baez on October 14, 2007 8:28 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 4)

Really enjoyable lectures, John. I’m looking forward to hearing some of Jim’s, too.

Before I make a mathematical comment, let me say that I had no trouble by simply clicking on the streaming video for the first lecture (received on the Atlantic side of the US), but for lecture number 4 I had to install an updated version of QuickTime where the streaming video did not work well at all (the downloaded version was fine). Whatever setup you used for the first lecture, I hope that can be an option for future broadcasts.

Regarding the discussion of the binomial coefficients (ordinary and $q$-) and their categorifications, there were quite a few machinations involved (what with all the cancellations of $q$’s and $q-1$’s) before you arrived at the final nice formula for the cardinality of the Grassmannian $Gr_{n, k}$ over $\mathbb{F}_q$. But that formula was suggestive of something which I don’t recall being brought out in discussion: from lecture 1 we know we can interpret categorified $n !_F$ as the variety of complete flags in $F^n$ (where the uncombed Young diagram is a vertical strip). So it would be natural to categorify the identity

$|Gr_{n, k}(\mathbb{F}_q)| = \frac{n !_q}{k !_q (n-k) !_q}$

as a fibration

$Flag(F^n) \to Gr_{n, k}(F)$

which takes a flag $X_0 \subset X_1 \subset \ldots \subset X_n = F^n$ to $X_k$, and where the fiber over the point $X_k$ is $Flag(X_k) \times Flag(X_n/X_k)$.

Then all those $q$ and $q-1$ factors are naturally interpreted in terms of cardinalities of Borel subgroups attached to these various Flag-factors (Borel subgroup = stabilizer of flag).

The multinomial coefficients have a similar categorification in terms of flag varieties.

Posted by: Todd Trimble on October 15, 2007 3:03 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 4)

Todd wrote:

Before I make a mathematical comment, let me say that I had no trouble by simply clicking on the streaming video for the first lecture (received on the Atlantic side of the US), but for lecture number 4 I had to install an updated version of QuickTime…

Right. The people at the Multimedia Center switched formats because they were having trouble with the first format. You probably don’t care, but the first format was obtained by recording the lecture and simultaneously uploading it via wireless to a server at UCR specially designed for this purpose. This worked great for lecture 1, but for later lectures the audio kept dropping out. Also, this format somehow made it difficult for them to offer the videos in downloadable form.

So, we’ve switched to recording the lectures on the camera and taking the camera over to the Multimedia Center after class, where they translate it from some proprietary Sony format to a .mov file, which is then made available both in streaming and audio form. But, this requires the viewers to have QuickTime.

And maybe there are other problems?

… the streaming video did not work well at all (the downloaded version was fine).

Can you say a bit about what was wrong with it? I had no trouble with it. Did you tell QuickTime what your bandwidth is? When I first fired it up, it checked that on its own. It claims to optimize its performance based on that information. Later it’s occasionally asked if that information has changed.

Whatever setup you used for the first lecture, I hope that can be an option for future broadcasts.

I sort of doubt it, unless lots of people report some consistent problem with the current format. So far Eric V. and Tim Silverman and I said it was fine; Jeff Morton reports that the video just stopped after a few seconds, and you report some unspecified problem. So, it’s not clear that anything is wrong, apart from the general tendency for the physical universe to do annoying things (which is why we went into math).

I’m looking forward to hearing some of Jim’s, too.

Me too! You can already watch one if you know where to look, but this is lecture 5 — the folks at the Multimedia Center still haven’t given me his previous lectures, lectures 2 and 3. So, I’m not advertising it, since most people will enjoy these things better if they watch them in some semblance of chronological order. You’re different, of course, since you helped invent all this stuff.

Nice math point! I wanted a slicker derivation of the $q$-binomial coefficients, but I didn’t see one, and I decided a little workout in counting was not a bad thing. When I write the perfect book on this subject (after I die), I’ll use your argument. Heck, I may even mention it in class.

In my lecture, I mentioned a couple cases where products in the combinatorial formulas correspond not to products of varieties but ‘twisted products’ — that is, fiber bundles. The total space of a bundle of varieties is the ‘twisted product’ of the base and the fiber — so its cardinality (for finite fields) or Euler characteristic (for $\mathbb{R}$ or $\mathbb{C}$) is just the product.

But, you’re noting that we can also think of the base as the total space ‘divided by’ the fiber.

I’d only been thinking of division in terms of ‘modding out by a transitive group action’.

Posted by: John Baez on October 15, 2007 7:48 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 4)

Can you say a bit about what was wrong with it? I had no trouble with it. Did you tell QuickTime what your bandwidth is? When I first fired it up, it checked that on its own.

On a purely descriptive level, it sounded like some very weird audio feedback, as if some evil John Baez doppelganger was tormenting you by talking over you and anticipating exactly what you were going to say in 30 seconds. I’ve no idea what was going on. QuickTime didn’t ask me what my bandwidth was, so I didn’t tell it. I assumed it checked for that itself, as it did in your case.

(Actually, using your tip, I’ve just clicked on streaming video for lecture 5 [Jim], and it seems to be working fine now.)

Posted by: Todd Trimble on October 15, 2007 8:24 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 4)

Okay, I’ll tell the folks at the Multimedia Center that an evil doppelganger is tormenting me.

Posted by: John Baez on October 15, 2007 8:47 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 4)

I like the concept of Geometric Representation Theory.

I certainly do not yet have a rigourus understanding. There appears to be a category flavor.

What type of advantages does this have over David Hestenes’ concept of Geometric Algebra / Calculus?

Posted by: Doug on October 15, 2007 2:28 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 4)

Doug wrote:

There appears to be a category flavor.

If you know category theory, everything has a category flavor. It’s a helpful framework for thinking about all sorts of things.

What type of advantages does this have over David Hestenes’ concept of Geometric Algebra / Calculus?

That’s a tough question — it’s a bit like asking what advantages a hammer has over a saw. They do very different things, and they should both be in your tool kit.

Geometric algebra uses Clifford algebras to study multivectors, differential forms and other linear-algebraic geometrical entities. Geometric representation theory builds linear representations of groups from group actions on sets.

Posted by: John Baez on October 15, 2007 3:03 AM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 4)

I’d like to mention how much the “category flavor”, or rather words like “categorified”, stand out in the new video format as opposed to previous seminars I’ve looked at.

I’m sure the same undercurrents have been there in the past, but there’s nothing like seeing it talked about and worked through live in simple (random!) examples to drum the concept home.

Seeing the familiar notation for categorified q-integers left me feeling I’d learned some aikido — through a barely perceived (notational) flick of the wrist, an 8 stone/112 lb/50 kg mathematical weakling like me feels he can suddenly throw heavily muscled ideas across the room! They normally just beat me up.

(the point being I guess that it’s hard to learn even beginners aikido from handwritten notes)

Posted by: Allan E on October 15, 2007 3:33 AM | Permalink | Reply to this

### Mackey functors and Geometric Representation Theory

I think something which is relevant to the Tale of Groupoidification is the thesis of Ross Street’s student Elango Panchadcharam, which I found through Google :

Elango Panchadcharam, Categories of Mackey Functors, Phd thesis Macquarie 2006.

I mentioned it before in this post.

In this work some interesting constructions can be found involving the following terms : finite $G$-sets, groupoids, spans, representations, Mackey functors, fully faithful embeddings.

It’s not really the same “direction” as the Tale of Groupoidification, but it should be noted nonetheless.

Posted by: Bruce Bartlett on October 15, 2007 6:30 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 4)

How much representing are you going to do? Is it just going to be of Weyl groups, or are you going to get as far as the Geometric Satake Correspondence?

Posted by: Allen Knutson on October 16, 2007 4:37 PM | Permalink | Reply to this

### Re: Geometric Representation Theory (Lecture 4)

Ideally I’m aiming to set Kazhdan–Lusztig theory, Khovanov homology, and the theory of quiver representations in a conceptually clear framework using groupoidification.

This would take a long time; the seminar will last at least 2 years, but I’m not sure how far we’ll get in that time. I need to introduce the whole theory of semisimple Lie algebras, Bruhat classes, quivers, Tits buildings and the like… but we decided it’s best to start by doing $A_n$-flavored stuff and then branch out. So, right now it’s all about $PGL(n)$ and $S_n$ (which we are calling $n!$).

Neither of us know geometric Satake well enough to have been wanting to add that to the list, but if we learn it I’m sure we’ll want to talk about it!

Most importantly, we want to have lots of fun and explain things in such simple terms that nobody in the universe can ever forget them afterwards.

Posted by: John Baez on October 16, 2007 7:59 PM | Permalink | Reply to this

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