It’s been a while, so let’s include a recap : a (transitive) permutation representation of the modular group is determined by the conjugacy class of a cofinite subgroup , or equivalently, to a dessin d’enfant. We have introduced a quiver (aka an oriented graph) which comes from a triangulation of the compactification of where is the hyperbolic upper half-plane. This quiver is independent of the chosen embedding of the dessin in the Dedeking tessellation. (For more on these terms and constructions, please consult the series Modular subgroups and Dessins d’enfants).

Why are quivers useful? To start, any quiver defines a noncommutative algebra, the path algebra , which has as a -basis all oriented paths in the quiver and multiplication is induced by concatenation of paths (when possible, or zero otherwise). Usually, it is quite hard to make actual computations in noncommutative algebras, but in the case of path algebras you can just see what happens.

Moreover, we can also see the finite dimensional representations of this algebra . Up to isomorphism they are all of the following form : at each vertex of the quiver one places a finite dimensional vectorspace and any arrow in the quiver determines a linear map between these vertex spaces, that is, to corresponds a matrix in . These matrices determine how the paths of length one act on the representation, longer paths act via multiplcation of matrices along the oriented path.

A necklace in the quiver is a closed oriented path in the quiver up to cyclic permutation of the arrows making up the cycle. That is, we are free to choose the start (and end) point of the cycle. For example, in the one-cycle quiver

the basic necklace can be represented as or or . How does a necklace act on a representation? Well, the matrix-multiplication of the matrices corresponding to the arrows gives a square matrix in each of the vertices in the cycle. Though the dimensions of this matrix may vary from vertex to vertex, what does not change (and hence is a property of the necklace rather than of the particular choice of cycle) is the trace of this matrix. That is, necklaces give complex-valued functions on representations of and by a result of Artin and Procesi there are enough of them to distinguish isoclasses of (semi)simple representations! That is, linear combinations a necklaces (aka super-potentials) can be viewed, after taking traces, as complex-valued functions on all representations (similar to character-functions).

In physics, one views these functions as potentials and it then interested in the points (representations) where this function is extremal (minimal) : the vacua. Clearly, this does not make much sense in the complex-case but is relevant when we look at the real-case (where we look at skew-Hermitian matrices rather than all matrices). A motivating example (the Yang-Mills potential) is given in Example 2.3.2 of Victor Ginzburg’s paper Calabi-Yau algebras.

Let be a super-potential (again, a linear combination of necklaces) then our commutative intuition tells us that extrema correspond to zeroes of all partial differentials where runs over all coordinates (in our case, the arrows of the quiver). One can make sense of differentials of necklaces (and super-potentials) as follows : the partial differential with respect to an arrow occurring in a term of is defined to be the path in the quiver one obtains by removing all 1-occurrences of in the necklaces (defining ) and rearranging terms to get a maximal broken necklace (using the cyclic property of necklaces). An example, for the cyclic quiver above let us take as super-potential (2 cyclic turns), then for example

(the first term corresponds to the first occurrence of , the second to the second). Okay, but then the vacua-representations will be the representations of the quotient-algebra (which I like to call the vacualgebra)

which in ‘physical relevant settings’ (whatever that means…) turn out to be Calabi-Yau algebras.

But, let us return to the case of subgroups of the modular group and their quivers. Do we have a natural super-potential in this case? Well yes, the quiver encoded a triangulation of the compactification of and if we choose an orientation it turns out that all ‘black’ triangles (with respect to the Dedekind tessellation) have their arrow-sides defining a necklace, whereas for the ‘white’ triangles the reverse orientation makes the arrow-sides into a necklace. Hence, it makes sense to look at the cubic superpotential being the sum over all triangle-sides-necklaces with a +1-coefficient for the black triangles and a -1-coefficient for the white ones. Let’s consider an index three example from a previous post

In this case the super-potential coming from the triangulation is

and therefore we have a noncommutative algebra associated to this index 3 subgroup. Contrary to what I believed at the start of this series, the algebras one obtains in this way from dessins d’enfants are far from being Calabi-Yau (in whatever definition). For example, using a GAP-program written by Raf Bocklandt Ive checked that the growth rate of the above algebra is similar to that of , so in this case can be viewed as a noncommutative curve (with singularities).

However, this is not the case for all such algebras. For example, the vacualgebra associated to the second index three subgroup (whose fundamental domain and quiver were depicted at the end of this post) has growth rate similar to that of …

I have an outlandish conjecture about the growth-behavior of all algebras coming from dessins d’enfants : the algebra sees what the monodromy representation of the dessin sees of the modular group (or of the third braid group). I can make this more precise, but perhaps it is wiser to calculate one or two further examples…

Last time (possibly with help from the survival guide) we have seen that the universal map from the modular group to its profinite completion (limit over all finite index normal subgroups ) gives an embedding of the sets of (continuous) simple finite dimensional representations

and based on the example we would like the above embedding to be dense in some kind of noncommutative analogon of the Zariski topology on .

We use the Zariski topology on as in these two M-geometry posts¹. So, what’s this idea in this special case? Let be the vectorspace with basis the conjugacy classes of elements of (that is, the space of class functions). As explained here it is a consequence of the Artin-Procesi theorem that the linear functions separate finite dimensional (semi)simple representations of . That is we have an embedding

and we can define closed subsets of as subsets of simple representations on which a set of class-functions vanish. With this definition of Zariski topology it is immediately clear that the image of is dense. For, suppose it would be contained in a proper closed subset then there would be a class-function vanishing on all simples of so, in particular, there should be a bound on the number of simples of finite quotients which clearly is not the case (just look at the quotients ).

But then, the same holds if we replace ’simples of ‘ by ’simple components of permutation representations of ‘. This is the importance of Farey symbols to the representation problem of the modular group. They give us a manageable subset of simples which is nevertheless dense in the whole space. To utilize this a natural idea might be to ask what such a permutation representation can see of the modular group, or in geometric terms, what the tangent space is to in a permutation representation². We will call this the modular content of the permutation representation and to understand it we will have to compute the tangent quiver .

1. already, I regret terminology, I should have just called it noncommutative geometry [↩]
2. more precisely, in the ‘cluster’ of points making up the simple components of the representation representation [↩]

For any finite dimensional A-representation S we defined before a character which is an linear functional on the noncommutative functions and defined via

for all

We would like to have enough such characters to separate simples, that is we would like to have an embedding

from the set of all finite dimensional simple A-representations into the linear dual of . This is a consequence of the celebrated Artin-Procesi theorem.

Michael Artin was the first person to approach representation theory via algebraic geometry and geometric invariant theory. In his 1969 classical paper “On Azumaya algebras and finite dimensional representations of rings” he introduced the affine scheme of all n-dimensional representations of A on which the group acts via basechange, the orbits of which are exactly the isomorphism classes of representations. He went on to use the Hilbert criterium in invariant theory to prove that the closed orbits for this action are exactly the isomorphism classes of semi-simple -dimensional representations. Invariant theory tells us that there are enough invariant polynomials to separate closed orbits, so we would be done if the caracters would generate the ring of invariant polynmials, a statement first conjectured in this paper.

Claudio Procesi was able to prove this conjecture in his 1976 paper “The invariant theory of matrices” in which he reformulated the fundamental theorems on -invariants to show that the ring of invariant polynomials of m matrices under simultaneous conjugation is generated by traces of words in the matrices (and even managed to limit the number of letters in the words required to ). Using the properties of the Reynolds operator in invariant theory it then follows that the same applies to the -action on the representation schemes .

So, let us reformulate their result a bit. Assume the affine -algebra A is generated by the elements then we define a necklace to be an equivalence class of words in the , where two words are equivalent iff they are the same upto cyclic permutation of letters. For example and determine the same necklace. Remark that traces of different words corresponding to the same necklace have the same value and that the noncommutative functions are spanned by necklaces.

The Artin-Procesi theorem then asserts that if S and T are non-isomorphic simple A-representations, then as elements of and even that they differ on a necklace in the generators of A of length at most . Phrased differently, the array of characters of simples evaluated at necklaces is a substitute for the clasical character-table in finite group theory.

This morning, Esther Beneish arxived the paper The center of the generic algebra of degree p that may contain the most significant advance in my favourite problem for over 15 years! In it she claims to prove that the center of the generic division algebra of degree p is stably rational for all prime values p. Let me begin by briefly explaining what the problem is all about. Consider one n by n matrix A which is sufficiently general, then it will have all its eigenvalues distinct, but then it is via the Jordan normal form theorem uniquely determined upto conjugation (that is, base change) by its characteristic polynomial. In other words, the conjugacy class of a sufficiently general n by n matrix depends freely on the coefficients of the characteristic polynomial (which are the n elementary symmetric functions in the eigenvalues of the matrix). Now what about couples of n by n matrices (A,B) under simultaneous conjugation (that is all couples of the form for some invertible n by n matrix g) ??? So, does there exist a sort of Jordan normal form for couples of n by n matrices which are sufficiently general? That is, are there a set of invariants for such couples which determine it is freely upto simultaneous conjugation?

For couples of 2 by 2 matrices, Claudio Procesi rediscovered an old result due to James Sylvester saying that this is indeed the case and that the set of invariants consists of the five invariants Tr(A),Tr(B),Det(A),Det(B) and Tr(AB). Now, Claudio did a lot more in his paper. He showed that if you could prove this for couples of matrices, you can also do it for triples, quadruples even any k-tuples of n by n matrices under simultaneous conjugation. He also related this problem to the center of the generic division algebra of degree n (which was introduced earlier by Shimshon Amitsur in a rather cryptic manner and for a while he simply refused to believe Claudio’s description of this division algebra as the one generated by two generic n by n matrices, that is matrices filled with independent variables). Claudio also gave the description of the center of this algebra as a field of lattice-invariants (over the symmetric group S(n) ) which was crucial in subsequent investigations. If you are interested in the history of this problem, its connections with Brauer group problems and invariant theory and a short description of the tricks used in proving the results I’ll mention below, you might have a look at the talk Centers of Generic Division Algebras, the rationality problem 1965-1990 I gave in Chicago in 1990.

The case of couples of 3 by 3 matrices was finally settled in 1979 by Ed Formanek and a year later he was able to solve also the case of couples of 4 by 4 matrices in a fabulous paper. In it, he used solvability of S(4) in an essential way thereby hinting at the possibility that the problem might no longer have an affirmative answer for larger values of n. When I read his 4×4 paper I believed that someone able to prove such a result must have an awesome insight in the inner workings of matrices and decided to dedicate myself to this problem the moment I would get a permanent job… . But even then it is a reckless thing to do. Spending all of your time to such a difficult problem can be frustrating as there is no guarantee you’ll ever write a paper. Sure, you can find translations of the problem and as all good problems it will have connections with other subjects such as moduli spaces of vectorbundles and of quiver representations, but to do the ‘next number’ is another matter.

Fortunately, early 1990, together with Christine Bessenrodt we were able to do the next two ‘prime cases’ : couples of 5 by 5 and couples of 7 by 7 matrices (Katsylo and Aidan Schofield had already proved that if you could do it for couples of k by k and l by l matrices and if k and l were coprime then you could also do it for couples of kl by kl matrices, so the n=6 case was already done). Or did we? Well not quite, our methods only allowed us to prove that the center is stably rational that is, it becomes rational by freely adjoining extra variables. There are examples known of stably rational fields which are NOT rational, but I guess most experts believe that in the case of matrix-invariants stable rationality will imply rationality. After this paper both Christine and myself decided to do other things as we believed we had reached the limits of what the lattice-method could do and we thought a new idea was required to go further. If today’s paper by Esther turns out to be correct, we were wrong. The next couple of days/weeks I’ll have a go at her paper but as my lattice-tricks are pretty rusty this may take longer than expected. Still, I see that in a couple of weeks there will be a meeting in Atlanta were Esther and all experts in the field will be present (among them David Saltman and Jean-Louis Colliot-Thelene) so we will know one way or the other pretty soon. I sincerely hope Esther’s proof will stand the test as she was the only one courageous enough to devote herself entirely to the problem, regardless of slow progress.

One way to increase the blogshare-value of this site might be to give readers more of what they want. In fact, there is an excellent guide for those who really want to increase traffic on their site called 26 Steps to 15k a Day. A somewhat sobering suggestion is rule S :

“Think about what people want. They aren't coming to your site to view “your content”, they are coming to your site looking for “their content”.”

But how do we know what people want? Well, by paying attention to Google-referrals according to rule U :

“The search engines will tell you exactly what they want to be fed - listen closely, there is gold in referral logs, it's just a matter of panning for it.”

And what do these Google-referrals show over the last couple of days? Well, here are the top recent key-words given to Google to get here :

13 : carolyn dean jacobian conjecture
11 : carolyn dean jacobian
9 : brauer severi varieties
7 : latexrender

7 : brauer severi
7 : spinor bundles
7 : ingalls azumaya
6 : [Unparseable or potentially dangerous latex formula Error 6 ]
6 : jacobian conjecture carolyn dean

See a pattern? People love to hear right now about the solution of the Jacobian conjecture in the plane by Carolyn Dean. Fortunately, there are a couple of things more I can say about this and it may take a while before you know why there is a photo of Tracy Chapman next to this post…

First, it seems I only got part of the Melvin Hochster email. Here is the final part I was unaware of (thanks to not even wrong)

Earlier papers established the following: if there is
a counterexample, the leading forms of $f$ and $g$ may
be assumed to have the form $(x^a y^b)^J$ and $(x^a y^b)^K$,
where $a$ and $b$ are relatively prime and neither $J$
nor $K$ divides the other (Abhyankar, 1977). It is known that
$a$ and $b$ cannot both be $1$ (Lang, 1991) and that one may
assume that $C[f,g]$ does not contain a degree one polynomial
in $x, y$ (Formanek, 1994).

Let $Dx$ and $Dy$ indicate partial differentiation with respect
to $x$ and $y$, respectively. A difficult result of Bass (1989)
asserts that if $D$ is a non-zero operator that is a polynomial
over $C$ in $x Dx$ and $y Dy$, $G$ is in $C[x,y]$ and $D(G)$
is in $C[f,g]$, then $G$ is in $C[f,g]$.

The proof proceeds by starting with $f$ and $g$ that give
a counterexample, and recursively constructing sequences of
elements and derivations with remarkable, intricate and
surprising relationships. Ultimately, a contradiction is
obtained by studying a sequence of positive integers associated
with the degrees of the elements constructed. One delicate
argument shows that the sequence is bounded. Another delicate
argument shows that it is not. Assuming the results described
above, the proof, while complicated, is remarkably self-contained
and can be understood with minimal background in algebra.

• Mel Hochster

Speaking about the Jacobian conjecture-post at not even wrong and the discussion in the comments to it : there were a few instances I really wanted to join in but I'll do it here. To begin, I was a bit surprised of the implicit attack in the post

Dean hasn't published any papers in almost 15 years and is nominally a lecturer in mathematics education at Michigan.

But this was immediately addressed and retracted in the comments :

Just curious. What exactly did you mean by “nominally a lecturer”?
Posted by mm at November 10, 2004 10:54 PM

I don't know anything about Carolyn Dean personally, just that one place on the Michigan web-site refers to her as a “lecturer”, another as a “visiting lecturer”. As I'm quite well aware from personal experience, these kinds of titles can refer to all sorts of different kinds of actual positions. So the title doesn't tell you much, which is what I was awkwardly expressing.
Posted by Peter at November 10, 2004 11:05 PM

Well, I know a few things about Carolyn Dean personally, the most relevant being that she is a very careful mathematician. I met her a while back (fall of 1985) at UCSD where she was finishing (or had finished) her Ph.D. If Lance Small's description of me would have been more reassuring, we might even have ended up sharing an apartment (quod non). Instead I ended up with Claudio Procesi… Anyway, it was a very enjoyable month with a group of young starting mathematicians and I fondly remember some dinner-parties we organized. The last news I heard about Carolyn was 10 to 15 years ago in Oberwolfach when it was rumoured that she had solved the Jacobian conjecture in the plane… As far as I recall, the method sketched by Hochster in his email was also the one back then. Unfortunately, at the time she still didn't have all pieces in place and a gap was found (was it by Toby Stafford? or was it Hochster?, I forgot). Anyway, she promptly acknowledged that there was a gap.
At the time I was dubious about the approach (mostly because I was secretly trying to solve it myself) but today my gut feeling is that she really did solve it. In recent years there have been significant advances in polynomial automorphisms (in particular the tame-wild problem) and in the study of the Hilbert scheme of points in the plane (which I always thought might lead to a proof) so perhaps some of these recent results did give Carolyn clues to finish off her old approach? I haven't seen one letter of the proof so I'm merely speculating here. Anyway, Hochster's assurance that the proof is correct is good enough for me right now.
Another discussion in the NotEvenWrong-comments was on the issue that several old problems were recently solved by people who devoted themselves for several years solely to that problem and didn't join the parade of dedicated follower of fashion-mathematicians.

It is remarkable that the last decade has seen great progress in math (Wiles proving Fermat's Last Theorem, Perelman proving the Poincare Conjecture, now Dean the Jacobian Conjecture), all achieved by people willing to spend 7 years or more focusing on a single problem. That's not the way academic research is generally structured, if you want grants, etc. you should be working on much shorter term projects. It's also remarkable that two out of three of these people didn't have a regular tenured position.

I think particle theory should learn from this. If some of the smarter people in the field would actually spend 7 years concentrating on one problem, the field might actually go somewhere instead of being dead in the water
Posted by Peter at November 13, 2004 08:56 AM

Here we come close to a major problem of today's mathematics. I have the feeling that far too few mathematicians dedicate themselves to problems in which they have a personal interest, independent of what the rest of the world might think about these problems. Far too many resort to doing trendy, technical mathematics merely because it is approved by so called 'better' mathematicians. Mind you, I admit that I did fall in that trap myself several times but lately I feel quite relieved to be doing just the things I like to do no matter what the rest may think about it. Here is a little bit of advice to some colleagues : get yourself an iPod and take some time to listen to songs like this one :

Don't be tempted by the shiny apple
Don't you eat of a bitter fruit
Hunger only for a taste of justice

Hunger only for a world of truth
'Cause all that you have is your soul

from Tracy Chapman's All that you have is your soul