Geometric Representation Theory (Lecture 20)

January 10, 2008

Geometric Representation Theory (Lecture 20)

Posted by John Baez

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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 hair-raising? Only time will tell!

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Lecture 20 (December 6) - John Baez on the Fundamental Theorem of Hecke Operators. This theorem says that for any finite group GG, if we take the Hecke bicategory of GG and ‘degroupoidify’ it using
$\overline{D}: [bicategories enriched over FinSpan] \to [categories enriched over FinVect]$

the result is equivalent to the category of (finite-dimensional) 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 q-deformed 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, Higher-Dimensional 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.

Posted at January 10, 2008 3:25 AM UTC

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Re: Geometric Representation Theory (Lecture 20)

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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.

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 2-category 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.

Posted by: Urs Schreiber on January 10, 2008 10:50 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 20)

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Urs wrote:

More people should do that.

When I was a grad student, I really enjoyed a course by Daniel Quillen where he was trying to develop a very simple proof of the Atiyah–Singer index theorem, using little besides elementary calculus and Clifford algebras. Every day he would start by reviewing, in an incredibly neat and well-organized way, what he’d done before. But, after the course got going, most classes would begin to ‘break down’ near the end, when he ran into material that he hadn’t fully developed yet, and sometimes hit problems.

So, we got to learn how someone actually does math research.

(Eventually his research program was ‘scooped’ by Ezra Getzler, who came up with a similar proof of the index theorem, but using more heavy-duty analysis than Quillen allowed himself.)

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.

That’s too bad, but I understand. I’ve heard from a secret source that the bureaucracy at your institution is so inefficient that they’ve never managed to give you a key to the front door. So, I’m not surprised they can’t get streaming videos to work.

I also understand why you’re so busy.

But, eventually I’ll put HDA7 on the arXiv, and by then it will cover a lot more interesting material than it does now — categorified quantum groups and the like. So, you can read that.

… $G_(n)//G_(n)$ : 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 groupoidification program.

It was still an interesting discussion. I raised my ever-burning question about your interest in groupoids of the form $G//G$ : why should you be interested in a groupoid that’s equivalent to the trivial groupoid? And, I think we worked our way a bit closer to the answer, which involved the ‘mapping cone’ concept. (In homotopy theory a mapping cone is contractible, hence ‘equivalent to the trivial space’, but the mapping cone construction is still useful.) You subsequently seem to have worked this idea into your bag of tricks… but I’m afraid I’m so busy that I don’t follow a lot of what you’re doing.

I think we had a useful interaction, even if we both scattered off and again behaved almost as free particles.

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.

That would be cool. The common attitude towards ‘geometric representation theory’ is that it studies linear representations of groups that arise from group actions on sets, algebraic varieties, etc. via various ‘linearization’ processes that turn these other entities into vector spaces: the vector space of functions on a set, the homology of a variety, etc. We’re trying to say that, at least in simple cases, the linearization process is almost an afterthought, and can be skipped if you get good enough at other kinds of math. We often use linear algebra just because we’re used to it.

It boils down to understanding, I guess, how exactly the 2-category of spans over $G$ is related to the category of linear $G$ -representations.

Well that’s good, because that’s what the Fundamental Theorem of Hecke Operators answers.

However, the correct answer involved a little bit of ‘unasking the question’!

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 thought I did state it, but maybe around 1 am while we were walking back to the hotel, somewhat lost in the streets of Vienna…

Anyway, I now understand all this stuff much better, and most of that understanding is built into the pompously named Fundamental Theorem of Hecke Algebras. The false version is easy to understand, as is the quick and dirty fix that makes it true… but the really good version is the content of HDA7 and is explained in simplified form in this lecture — lecture 20.

In a nutshell: degroupoidifying the Hecke bicategory of a finite group $G$ , we get the category of finite-dimensional permutation representations of $G$ . And, not explained above: if we then split idempotents, we get the category of all finite-dimensional representations of $G$ .

(The limitation to finite groups is to avoid certain technical issues that make the general theory a lot more complicated and/or interesting.)

Posted by: John Baez on January 10, 2008 9:46 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 20)

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why should you be interested in a groupoid that’s equivalent to the trivial groupoid?

The mapping cone concept was useful for orienting ourselves, but I think the real answer is something which I may still not be able to formulate, but which a sufficiently sophisticated person should be able to formulate after thinking about the answer to the analogous question:

Why should we be interesting in the space $E G$ ? After all, it is equivalent to a point!

In fact, this question is considerably more than analogous: $G // G$ is $E G$ , in a sense.

This sense is: the sequence of groupoids

$G \to G // G \to \mathbf{B}G$

is mapped by the nerve realization functor to the universal $G$ -bundle

$G \to E G \to B G \,.$

While the middle term in both cases is equivalent to something trivial, I guess it is the fact that it sits in this sequence which makes it interesting.

So far so good. The really striking additional point now is that when we think of bundles with connection, $G // G$ is the home of curvature.

So, while you might have to further help me with formulating the abstract reason for “why is $G // G$ interesting?”, I can hand you large amounts of experimental evidence that indeed it is.

When $G$ is Lie, hitting $G \to G // G \to \mathbf{B}G$ with something like a functor from Lie groups to dg-algebras produces

$CE(g) \leftarrow \mathrm{W}(g) \leftarrow inv(g) \,.$

You can browse through our Lie $\infty$ -connections and see $\mathrm{W}(g)$ appear all over the place.

You can imagine that there is an integration procedure so that whenever you see a $\mathrm{W}(g)$ , it turns into $\mathbf{B}(G // G)$ (and whenever you see a $\mathrm{CE}(g)$ it turns into $\mathbf{B} G$ (for $G$ a Lie $n$ -group now).

This is just to show: $G // G$ appears all over the place, and it is extremely useful.

Figure 1 on p. 6 illustrates the above analogy.

What is it that makes it useful, even though it is contractible?

It’s the fact that we know what “vertical” and what “horizontal” is inside $G // G$ , and if we arrange that to be respected, then we prevent $G // G$ from collapsing to a point.

You see this general idea appearing first in figure 3 on p. 25. Then in figure 8 on p. 46 it appears in the context of $n$ -transport, where finally in the diagram of the crucial proposition 25 on p. 48 it achieves its full meaning: that diagram, in a visually obvious sense says:

stuff moving vertically in $G // G$ does not affect horizontal stuff in $G // G$ .

And that’s a very important statement, which only the existence of $G // G$ allows us to state efficiently.

So in words, the crucial insight is: $n$ -transport and its curvature take values in $G // G$ . The fact that this does not make them trivial is that they are constrained to “move” only “vertically” in $G // G$ .

And that constraint, by the way, also has a nice interpretation: the thing taking values in $G//G$ arises really in an extension provlem as the obstruction to in a way trivializing the vertical part. That’s my best current attempt at giving an “abstract explanation” for why we should care about $G // G$ .

Anyway, my point here is this: there is no doubt that we need to think of $G$ $n$ -transport as really being $G // G$ ( $n+1$ )-transport plus constraints. That’s one thing the Lie $\infty$ -connection work establishes.

So I find myself with a transport that assigns $G // G$ to fibers – and then I pass to the $n$ -Café and read that you are teaching that this is a sneaky way to look at $G$ -vector spaces.

That makes me wonder.

Posted by: Urs Schreiber on January 10, 2008 11:35 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 20)

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John wrote:

In a nutshell: degroupoidifying the Hecke bicategory of a finite group $G$ , we get the category of finite-dimensional permutation representations of $G$ . And, not explained above: if we then split idempotents, we get the category of all finite-dimensional representations of $G$ .

Aargh! The second sentence is only true for certain groups, like the permutation groups $n!$ . In general things are trickier, because not all irreps of $G$ appear as subrepresentations of permutation representations.

Double aargh! What an idiot! The second sentence up there is actually true for all finite groups $G$ , if we work over the complex numbers. Every irreps of $G$ appears as a subrepresentation of a permutation representation — namely, the regular representation.

There are some other things that only work for special groups, but this is completely general: for any finite group $G$ , we can get the category of finite-dimensional complex representations by taking the Hecke bicategory of $G$ , degroupoidifying, and splitting idempotents.

Posted by: John Baez on January 30, 2008 4:37 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 20)

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John wrote:

In a nutshell: degroupoidifying the Hecke bicategory of a finite group $G$ , we get the category of finite-dimensional permutation representations of $G$ . And, not explained above: if we then split idempotents, we get the category of all finite-dimensional representations of $G$ .

Aargh! The second sentence is only true for certain groups, like the permutation groups $n!$ . In general things are trickier, because not all irreps of $G$ appear as subrepresentations of permutation representations. We’ve already discussed this issue. To get all complex reps of these other groups it seems we need to introduce $\mathbb{C}$ ‘by hand’ — or at least the field generated by all roots of unity, $\mathbb{Q}^{ab}$ .

Posted by: John Baez on January 11, 2008 6:58 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 20)

I am not very knowledgeable in representation theory, but I’m curious about something and it’s related to some of the previous discussion. The question is very straightforward.

Given S_n, the symmetric group on n elements, define the permutation representation R defined by R(g) = [g], where [g] is the permutation matrix corresponding to the permutation g. What is known about the irreducible subrepresentations of R? In particular, I would like to know the sum of dimensions of the distinct irreducible subrepresentations of R. I’d appreciate if you could provide me with a reference. Thanks in advance!

Posted by: Arnab on February 3, 2008 11:22 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 20)

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Arnab wrote:

Given $S_n$ , the symmetric group on $n$ elements, define the permutation representation $R$ defined by $R(g) = [g]$ , where $[g]$ is the permutation matrix corresponding to the permutation $g$ . What is known about the irreducible subrepresentations of $R$ ?

It has a 1-dimensional irreducible subrepresentation consisting of all vectors of the form $(x,x,\dots, x)$ . This is just the trivial representation. The orthogonal complement of this subspace is an $(n-1)$ -dimensional irreducible representation.

So, $R$ breaks up into exactly two irreducible pieces (if $n \ge 2$ ).

This fact is a special case of a theorem Jim mentioned in lecture 7: if a finite group $G$ acts in a doubly transitive way on an $n$ -element set, the resulting representation of $G$ on the vector space $\mathbb{C}^n$ is a direct sum of the trivial 1-dimensional representation and an irreducible $(n-1)$ -dimensional representation. Jim sketched the proof, which is a nice application of the ideas we’re studying.

Posted by: John Baez on February 4, 2008 6:57 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 20)

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How much structure is needed to do the transfer

$H_0(X) \leftarrow H_0(Y) : f^!$

of groupoid 0-homologies given a functor $f : X \to Y$

of groupoids. I am asking because I would like to know if this can be understood at the place where the concept of a field enters, maybe.

Namely, when you form 0-th cohomology of a groupoid, you get a set, and it seems to be completely pointless overhead, at that point, to instead of talking about that set to talk about the vector space spanned by it.

But not so for the transfer. It is here that we need to use multiplication and division and addition, in order for the thing even to exist.

This is clear, but what I would like to know is: can we maybe understand the passage from sets to the vector spaces over fields which they span as something like the minimal prerequissite in order for there to be a transfer for the pushforward of homology?

Do you see what I mean?

Posted by: Urs Schreiber on January 10, 2008 9:15 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 20)

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Urs wrote:

How much structure is needed to do the transfer

$H_0(X) \leftarrow H_0(Y) : f^!$

of groupoid 0-homologies given a functor $f : X \to Y$

of groupoids?

No extra structure — just extra properties!

If the groupoids are finite, any functor between them gives rise to a transfer map on homology, using the simple formula given in HDA7. If the groupoids are infinite, the sum in this formula might diverge, and the cardinalities in this formula might be infinite. But, there are many interesting cases where those problems don’t occur… and you might say the winter’s seminar will be all about those cases.

Posted by: John Baez on January 10, 2008 9:55 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 20)

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There is a typo in Lemma 11 on p. 5 of the HDA VII draft: it says $X^{Set}$ as opposed to $Set^X$ .

Posted by: Urs Schreiber on January 10, 2008 9:36 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 20)

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Thanks — fixed!

Posted by: John Baez on January 11, 2008 8:15 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 20)

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Could that material covered by Mark Weber in strict 2-toposes be useful? So that instead of functors between $Set^X$ and $Set^Y$ , you think of them as going between discrete opfibrations over $X$ to the same over $Y$ ?

Posted by: David Corfield on January 11, 2008 12:11 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 20)

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Hey, isn’t the weak quotient $X // G$ , for a groupoid $G$ acting on a set $X$ just the pullback over Mark Weber’s classifying discrete opfibration in the 2-topos of categories, Pointed set $\to$ Set?

So $X // G$ projects down to $G$ , arrows $(x, g)$ being sent to $g$ .

$X // G$ also gets mapped to Pointed Set, object $x$ being sent to $(X, x)$ .

Posted by: David Corfield on January 12, 2008 9:51 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 20)

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Hey, isn’t the weak quotient $X//G$ , for a groupoid $G$ acting on a set $X$ just the pullback over Mark Weber’s classifying discrete opfibration in the 2-topos of categories, $Pointed set \to Set$ ?

Yes indeed. $X//G$ is another name for the category of elements of the functor $X: G \to Set$ , and this category of elements is obtained by pulling back the universal category of elements, $Pointed set \to Set$ , along $X$ . Exactly as you say.

Another useful thing to remember in this game is that

$Set^{X//G} \simeq Set^G/X.$

This is actually a general piece of abstract nonsense: that a slice of a presheaf topos $Set^C/X$ is (equivalent to) a presheaf topos, namely

$Set^{C \darr X}$

where the exponent denotes a comma category. This comma category is the category of elements of $X$ , which brings us back to what is usually written as $X//G$ when $C = G$ is a groupoid.

Posted by: Todd Trimble on January 12, 2008 12:05 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 20)

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A variance mistake may have crept into my previous comment: I have the feeling that the correct formulation should have been

$Set^{C^{op}}/X \simeq Set^{(C \darr X)^{op}}.$

Although for groupoids the mistake is fairly harmless :-).

While I’m at it, I may as well mention one more useful equivalence (it may well have been said earlier, but this blog moves fast and I don’t read everything; plus, I’m way behind on my video-watching):

$Set^G/(G/H) \simeq Set^{(G/H)//G)} \simeq Set^H.$

Posted by: Todd Trimble on January 12, 2008 2:26 PM | Permalink | Reply to this

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Re: Geometric Representation Theory (Lecture 20)

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What would happen to the groupoidification program if I were to decide that the spans appearing there are really telling me that we are secretly working in a homotopy category?

In that case i would discard the morphisms between spans and instead require the left leg of each span to be a weak equivalence. Or rather, replace single spans with sequences of spans with that property.

Would the main theorem relating this setup to linear representation theory still go through?

I am wondering what would happen if one looked at $Ho(\omega Grpd)$ , using this model category structure.

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