Geometric Representation Theory (Lecture 20)
Posted by John Baez
In this, the final lecture of the fall’s Geometric Representation Theory seminar, I tried to wrap up by giving a correct statement of the Fundamental Theorem of Hecke Operators.
The fall seminar was a lot of fun, and very useful. It didn’t go the way I expected. I thought I thoroughly understood groupoidification, but I didn’t! So, all hell broke loose when I tried to state the Fundamental Theorem. The seminar threatened to swerve out of control, and Jim had to invent some more math to save the day. We skidded to safety at the very last second… but in the process, we learned a lot.
Will next quarter’s seminar be less hairraising? Only time will tell!

Lecture 20 (December 6)  John Baez on the Fundamental Theorem of Hecke Operators.
This theorem says that for any finite group $G$, if we take the Hecke bicategory of $G$ and
‘degroupoidify’ it using
$\overline{D}: [bicategories enriched over FinSpan] \to [categories enriched over FinVect]$
the result is equivalent to the category of (finitedimensional) permutation representations of $G$.
In short: the Hecke bicategory of $G$ is a groupoidification of the the category of permutation representations of $G$.
Future directions: groupoidifying the qdeformed Pascal’s triangle, the action of the quantum group $GL_q(2)$ on the quantum plane, and more generally the action of $GL_q(n)$ on ‘quantum $n$space’. (Final words cut off as the power cable to the video camera is accidentally unplugged!)
For a more precise and thorough statement of the Fundamental Theorem, read this:
 Supplementary reading: John Baez, HigherDimensional Algebra VII: Groupoidification (draft version).

Streaming
video in QuickTime format; the URL is
http://mainstream.ucr.edu/baez_12_06_stream.mov  Downloadable video
 Lecture notes by Alex Hoffnung
 Lecture notes by Apoorva Khare
To save time, I cut some corners in stating the Fundamental Theorem in class. The really beautiful statement involves a bit of topos theory. There’s more about this in the “supplementary reading”, but the technical details have not yet been optimized. Just as we can talk about group actions, we can talk about groupoid actions: an action of a groupoid $X$ is just a functor
$F: X \to Set$
If we know the all the actions of a groupoid — or more precisely, the topos of all its actions — we can recover the groupoid. But, Jim and I didn’t find a good constructive recipe for doing this. There are some other loose ends, too.
Luckily, Tom Leinster and Todd Trimble have been helping me out — and with such assistance, victory is inevitable.
Re: Geometric Representation Theory (Lecture 20)
This is really moving. I am glad that you do things like that – and do it in public. More people should do that.
I have to admit that while I was initially quivering with anticipation to learn more about the groupoidification program, when your seminar really picked up steam I found myself busy with such a bunch of other things that I gave up on trying to follow. Plus, I can’t easily watch your videos here in my office for stupid technical reasons, so I didn’t actually watch any one of them.
So I am glad to see HDA VII come into existence!
Just in my last entry I again went on about how striking it is that when we look at principal $n$bundles with connection and do the $n$curvature part right, we are looking at a bundle whose fibers don’t look like $G_{(n)}$, the structure $n$group, but like $G_{(n)}// G_{(n)}$: the action groupoid of that thing on itself.
I mentioned that a couple of times when we talked in Vienna last time, but probably failed to infect you with my excitement about this fact in the light of your groupoiification program.
And of course I might be hallucinating. But it seriously looks to me as if this is telling us that “geometric representations” (as opposed to linear representations, is that how you use the term?) arise automatically in $n$transport, and that hence maybe the usual subsequent passage to associated linear $n$transport is not real.
If and when I dig more into the groupoidification program, this will be the question I want to find the answer to.
It boils down to understanding, I guess, how exactly the 2category of spans over $G$ is related to the category of linear $G$representations.
I seem to recall that when we last talked about that in Vienna, you indicated a bunch of nice relationships, but didn’t quite come down to stating the entire theorem. Is that right? I might be misremembering. In any case, this is what I would like to understand.