The black&white psychedelic picture on the left of a tessellation of the hyperbolic upper-halfplane, was called the Dedekind tessellation in this post, following the reference given by John Stillwell in his excellent paper Modular Miracles, The American Mathematical Monthly, 108 (2001) 70-76.

But is this correct terminology? Nobody else uses it apparently. So, let’s try to track down the earliest depiction of this tessellation in the literature…

Stillwell refers to Richard Dedekind’s 1877 paper “Schreiben an Herrn Borchard uber die Theorie der elliptische Modulfunktionen”, which appeared beginning of september 1877 in Crelle’s journal (Journal fur die reine und angewandte Mathematik, Bd. 83, 265-292).

There are a few odd things about this paper. To start, it really is the transcript of a (lengthy) letter to Herrn Borchardt (at first, I misread the recipient as Herrn Borcherds which would be really weird…), written on June 12th 1877, just 2 and a half months before it appeared… Even today in the age of camera-ready-copy it would probably take longer.

There isn’t a single figure in the paper, but, it is almost impossible to follow Dedekind’s arguments without having a mental image of the tessellation. He gives a fundamental domain for the action of the modular group on the hyperbolic upper-half plane (a fact already known to Gauss) and goes on in section 3 to give a one-to-one mapping between this domain and the complex plane using what he calls the ‘valenz’ function (which is our modular function , making an appearance in moonshine, and responsible for the black&white tessellation, the two colours corresponding to pre-images of the upper or lower half-planes).

Then there is this remarkable opening sentence.

Sie haben mich aufgefordert, eine etwas ausfuhrlichere Darstellung der Untersuchungen auszuarbeiten, von welchen ich, durch das Erscheinen der Abhandlung von Fuchs veranlasst, mir neulich erlaubt habe Ihnen eine kurze Ubersicht mitzuteilen; indem ich Ihrer Einladung hiermit Folge leiste, beschranke ich mich im wesentlichen auf den Teil dieser Untersuchungen, welcher mit der eben genannten Abhandlung zusammenhangt, und ich bitte Sie auch, die Ubergehung einiger Nebenpunkte entschuldigen zu wollen, da es mir im Augenblick an Zeit fehlt, alle Einzelheiten auszufuhren.

Well, just try to get a paper (let alone a letter) accepted by Crelle’s Journal with an opening line like : “I’ll restrict to just a few of the things I know, and even then, I cannot be bothered to fill in details as I don’t have the time to do so right now!” But somehow, Dedekind got away with it.

So, who was this guy Borchardt? How could this paper be published so swiftly? And, what might explain this extreme ‘je m’en fous’-opening ?

Carl Borchardt was a Berlin mathematician whose main claim to fame seems to be that he succeeded Crelle in 1856 as main editor of the ‘Journal fur reine und…’ until 1880 (so in 1877 he was still in charge, explaining the swift publication). It seems that during this time the ‘Journal’ was often referred to as “Borchardt’s Journal” or in France as “Journal de M Borchardt”. After Borchardt’s death, the Journal für die Reine und Angewandte Mathematik again became known as Crelle’s Journal.

As to the opening sentence, I have a toy-theory of what was going on. In 1877 a bitter dispute was raging between Kronecker (an editor for the Journal and an important one as he was the one succeeding Borchardt when he died in 1880) and Cantor. Cantor had published most of his papers at Crelle and submitted his latest find : there is a one-to-one correspondence between points in the unit interval [0,1] and points of d-dimensional space! Kronecker did everything in his power to stop that paper to the extend that Cantor wanted to retract it and submit it elsewhere. Dedekind supported Cantor and convinced him not to retract the paper and used his influence to have the paper published in Crelle in 1878. Cantor greatly resented Kronecker’s opposition to his work and never submitted any further papers to Crelle’s Journal.

Clearly, Borchardt was involved in the dispute and it is plausible that he ‘invited’ Dedekind to submit a paper on his old results in the process. As a further peace offering, Dedekind included a few ‘nice’ words for Kronecker

Bei meiner Versuchen, tiefer in diese mir unentbehrliche Theorie einzudringen und mir einen einfachen Weg zu den ausgezeichnet schonen Resultaten von Kronecker zu bahnen, die leider noch immer so schwer zuganglich sind, enkannte ich sogleich…

Probably, Dedekind was referring to Kronecker’s relation between class groups of quadratic imaginary fields and the j-function, see the miracle of 163. As an added bonus, Dedekind was elected to the Berlin academy in 1880…

Anyhow, no visible sign of ‘Dedekind’s’ tessellation in the 1877 Dedekind paper, so, we have to look further. I’m fairly certain to have found the earliest depiction of the black&white tessellation (if you have better info, please drop a line). Here it is

It is figure 7 in Felix Klein’s paper “Uber die Transformation der elliptischen Funktionen und die Auflosung der Gleichungen funften Grades” which appeared in may 1878 in the Mathematische Annalen (Bd. 14 1878/79). He even adds the j-values which make it clear why black triangles should be oriented counter-clockwise and white triangles clockwise. If Klein would still be around today, I’m certain he’d be a metapost-guru.

So, perhaps the tessellation should be called Klein’s tessellation?? Well, not quite. Here’s what Klein writes wrt. figure 7

Diese Figur nun - welche die eigentliche Grundlage fur das Nachfolgende abgibt - ist eben diejenige, von der Dedekind bei seiner Darstellung ausgeht. Er kommt zu ihr durch rein arithmetische Betrachtung.

Case closed : Klein clearly acknowledges that Dedekind did have this picture in mind when writing his 1877 paper!

But then, there are a few odd things about Klein’s paper too, and, I do have a toy-theory about this as well… (tbc)

Bruce Westbury has a page on recent work on series of Lie groups including exceptional groups. Moreover, he did put his slides of a recent talk (probably at MPI) online.

Probably, someone considered a similar problem for simple groups. Are there natural constructions leading to a series of finite simple groups including some sporadic groups as special members ? In particular, does the following sequence appear somewhere ?

Here, is the simple group of order 168 (the automorphism group of the Klein quartic), and are the sporadic Mathieu groups and the are the alternating simple groups.

I’ve stumbled upon this series playing around with Farey sequences and their associated ‘dessins d’enfants’ (I’ll come back to the details of the construction another time) and have dubbed this sequence the Iguanodon series because the shape of the doodle leading to its first few terms

reminded me of the Iguanodons of Bernissart (btw. this sketch outlines the construction to the experts). Conjecturally, all groups appearing in this sequence are simple and probably all of them (except for the first few) will be alternating.

I did verify that none of the known low-dimensional permutation representations of other sporadic groups appear in the series. However, there are plenty of similar sequences one can construct from the Farey sequences, and it would be nice if one of them would contain the Conway group . (to be continued)

Here a list of saved pdf-files of previous NeverEndingBooks-posts on geometry in reverse chronological order.

(more…)

Today we will explain how curves defined over determine permutation representations of the carthographic groups. We have seen that any smooth projective curve (a Riemann surface) defined over the algebraic closure of the rationals, defines a Belyi map which is only ramified over the three points . By this we mean that there are exactly points of lying over any other point of (we call the degree of ) and that the number of points over and is smaller than . To such a map we associate a dessin d\’enfant, a drawing on linking the pre-images of and with exactly edges (the preimages of the open unit-interval). Next, we look at the preimages of and associate a permutation of letters to it by cycling counter-clockwise around these preimages and recording the edges we meet. We repeat this procedure for the preimages of and get another permutation . That is, we obtain a subgroup of the symmetric group which is called the monodromy group of the covering .

For example, the dessin on the right is associated to a degree map and if we let the black (resp. starred) vertices be the preimages of (respectively of ), then the corresponding partitions are and and the monodromy group is the alternating group (use GAP ).

But wait! The map is also ramified in so why don\’t we record also a permutation and are able to compute it from the dessin? (Note that all three partitions are needed if we want to reconstruct from the sheets as they encode in which order the sheets fit together around the preimages). Well, the monodromy group of a covering ramified only in three points is an epimorphic image of the fundamental group of the sphere minus three points That is, the group of all loops beginning and ending in a basepoint upto homotopy (that is, two such loops are the same if they can be transformed into each other in a continuous way while avoiding the three points).

This group is generated by loops running from the basepoint to nearby the i-th point, doing a counter-clockwise walk around it and going back to be basepoint and the epimorphism to the monodromy group is given by sending

Now, these three generators are not independent. In fact, this fundamental group is

To understand this, let us begin with an easier case, that of the sphere minus one point. The fundamental group of the plane minus one point is as it encodes how many times we walk around the point. However, on the sphere the situation is different as we can make our walk around the point longer and longer until the whole walk is done at the backside of the sphere and then we can just contract our walk to the basepoint. So, there is just one type of walk on a sphere minus one point (upto homotopy) whence this fundamental group is trivial. Next, let us consider the sphere minus two points

Repeat the foregoing to the walk , that is, strech the upper part of the circular tour all over the backside of the sphere and then we see that we can move it to fit with the walk $\sigma1$ BUT for the orientation of the walk! That is, if we do this modified walk we just made the trivial walk. So, this fundamental group is This is also the proof of the above claim. For, we can modify the third walk continuously so that it becomes the walk but with the reversed orientation ! As this allows us to compute the \’missing\’ permutation In the example above, we obtain so it has two cycles corresponding to the fact that the dessin has two regions (remember we should draw ths on the sphere) : the head and the outer-region. Hence, the pre-images of $\infty$ correspond to the different regions of the dessin on the curve . For another example, consider the degree 168 map

which is the modified orbit map for the action of on the Klein quartic. The corresponding dessin is the heptagonal construction of the Klein quartic

Here, the pre-images of 1 correspond to the midpoints of the 84 edges of the polytope whereas the pre-images of 0 correspond to the 56 vertices. We can label the 168 half-edges by numbers such that and are the standard generators b resp. a of the 168-dimensional regular representation (see the atlas page ). Calculating with GAP the element one finds that this permutation consists of 24 cycles of length 7, so again, the pre-images of lie one in each of the 24 heptagonal regions of the Klein quartic. Now, we are in a position to relate curves defined over via their Belyi-maps and corresponding dessins to Grothendiecks carthographic groups , and . The dessin gives a permutation representation of the monodromy group and because the fundamental group of the sphere minus three points is the free group op two generators, we see that any dessin determines a permutation representation of the congruence subgroup (see this post where we proved that this group is free). A clean dessin is one for which one type of vertex has all its valancies (the number of edges in the dessin meeting the vertex) equal to one or two. (for example, the pre-images of 1 in the Klein quartic-dessin or the pre-images of 1 in the monsieur Mathieu example ) The corresponding permutation then consists of 2-cycles and hence the monodromy group gives a permutation representation of the free product Finally, a clean dessin is said to be a quilt dessin if also the other type of vertex has all its valancies equal to one or three (as in the Klein quartic or Mathieu examples). Then, the corresponding permutation has order 3 and for these quilt-dessins the monodromy group gives a permutation representation of the free product Next time we will see how this lead Grothendieck to his anabelian geometric approach to the absolute Galois group.

Last time we have seen that the noncommutative manifold of a Riemann surface can be viewed as that Riemann surface together with a loop in each point. The extra loop-structure tells us that all finite dimensional representations of the coordinate ring can be found by separating over points and those living at just one point are classified by the isoclasses of nilpotent matrices, that is are parametrized by the partitions (corresponding to the sizes of the Jordan blocks). In addition, these loops tell us that the Riemann surface locally looks like a Riemann sphere, so an equivalent mental picture of the local structure of this noncommutative manifold is given by the picture on teh left, where the surface is part of the Riemann surface and a sphere is placed at every point. Today we will consider genuine noncommutative curves and describe their corresponding noncommutative manifolds.

Here, a mental picture of such a noncommutative sphere to keep in mind would be something like the picture on the right. That is, in most points of the sphere we place as before again a Riemann sphere but in a finite number of points a different phenomen occurs : we get a cluster of infinitesimally nearby points. We will explain this picture with an easy example. Consider the complex plane , the points of which are just the one-dimensional representations of the polynomial algebra in one variable (any algebra map is fully determined by the image of z). On this plane we have an automorphism of order two sending a complex number z to its negative -z (so this automorphism can be seen as a point-reflexion with center the zero element 0). This automorphism extends to the polynomial algebra, again induced by sending z to -z. That is, the image of a polynomial under this automorphism is f(-z).

With this data we can form a noncommutative algebra, the skew-group algebra the elements of which are either of the form or where is the cyclic group of order two generated by the automorphism g and f(z),g(z) are arbitrary polynomials in z.

The multiplication on this algebra is determined by the following rules

whereas

whereas

That is, multiplication in the factor is the usual multiplication, multiplication in the factor is the usual group-multiplication but when we want to get a polynomial from right to left over a group-element we have to apply the corresponding automorphism to the polynomial (thats why we call it a _skew group-algebra).

Alternatively, remark that as a -algebra the skew-group algebra is an algebra with unit element 1 = 1\aste and is generated by the elements and and that the defining relations of the multiplication are

and

hence another description would be

It can be shown that skew-group algebras over the coordinate ring of smooth curves are noncommutative smooth algebras whence there is a noncommutative manifold associated to them. Recall from last time the noncommutative manifold of a smooth algebra A is a device to classify all finite dimensional representations of A upto isomorphism Let us therefore try to determine some of these representations, starting with the one-dimensional ones, that is, algebra maps from

Such a map is determined by the image of X and that of Y. Now, as we have just two choices for the image of Y namely +1 or -1. But then, as the image is a commutative algebra and as XY+YX=0 we must have that the image of 2XY is zero whence the image of X must be zero. That is, we have only two one-dimensional representations, namely and

This is odd! Can it be that our noncommutative manifold has just 2 points? Of course not. In fact, these two points are the exceptional ones giving us a cluster of nearby points (see below) whereas most points of our noncommutative manifold will correspond to 2-dimensional representations!

So, let’s hunt them down. The center of (that is, the elements commuting with all others) consists of all elements of the form with f an _even polynomial, that is, f(z)=f(-z) (because it has to commute with 1\ast g), so is equal to the subalgebra .

The manifold corresponding to this subring is again the complex plane of which the points correspond to all one-dimensional representations of (determined by the image of ).

We will now show that to each point of corresponds a simple 2-dimensional representation of .

If a is not zero, we will consider the quotient of the skew-group algebra modulo the twosided ideal generated by . It turns out that

where the skew-group algebra on the right is given by the automorphism g on interchanging the two factors. If you want to become more familiar with working in skew-group algebras work out the details of the fact that there is an algebra-isomorphism between and the algebra of matrices . Here is the identification

so you have to verify that multiplication on the left hand side (that is in ) coincides with matrix-multiplication of the associated matrices.

Okay, this begins to look like what we are after. To every point of the complex plane minus zero (or to every point of the Riemann sphere minus the two points ) we have associated a two-dimensional simple representation of the skew-group algebra (btw. simple means that the matrices determined by the images of X and Y generate the whole matrix-algebra).

In fact, we now have already classified ‘most’ of the finite dimensional representations of , namely those n-dimensional representations

for which the image of X is an invertible matrix. We can show that such representations only exist when n is an even number, say n=2m and that any such representation is again determined by the geometric/combinatorial data we found last time for a Riemann surface.

That is, It is determined by a finite number of points from where k is at most m. For each index i we have a positive number such that and finally for each i we also have a partition of .

That is our noncommutative manifold looks like all points of with one loop in each point. However, we have to remember that each point now determines a simple 2-dimensional representation and that in order to get all finite dimensional representations with det(X) non-zero we have to scale up representations of by a factor two. The technical term here is that of a Morita equivalence (or that the noncommutative algebra is an Azumaya algebra over ).

What about the remaining representations, that is, those for which Det(X)=0? We have already seen that there are two 1-dimensional representations and lying over 0, so how do they fit in our noncommutative manifold? Should we consider them as two points and draw also a loop in each of them or do we have to do something different? Rememer that drawing a loop means in our geometry -> representation dictionary that the representations living at that point are classified in the same way as nilpotent matrices.

Hence, drawing a loop in would mean that we have a 2-dimensional representation of (different from ) and any such representation must correspond to matrices

and

But this is not possible as these matrices do not satisfy the relation XY+YX=0. Hence, there is no loop in and similarly also no loop in .

However, there are non semi-simple two dimensional representations build out of the simples and . For, consider the matrices

and

then these matrices do satisfy XY+YX=0! (and there is another matrix-pair interchanging in the Y-matrix). In erudite terminology this says that there is a nontrivial extension between and and one between and .

In our dictionary we will encode this information by the picture

where the two vertices correspond to the points and and the arrows represent the observed extensions. In fact, this data suffices to finish our classification project of finite dimensional representations of the noncommutative curve .

Those with Det(X)=0 are of the form : where R is a representation with invertible X-matrix (which we classified before) and T is a direct sum of representations involving only the simple factors and and obtained by iterating the 2-dimensional idea. That is, for each factor the Y-matrix has alternating along the diagonal and the X-matrix is the full nilpotent Jordan-matrix.

So here is our picture of the __noncommutative manifold of the noncommutative curve _ : the points are all points of together with one loop in each of them together with two points lying over 0 where we draw the above picture of arrows between them. One should view these two points as lying infinetesimally close to each other and the gluing data

contains enough information to determine that all other points of the noncommutative manifold in the vicinity of this cluster should be two dimensional simples! The methods used in this simple minded example are strong enough to determine the structure of the noncommutative manifold of any noncommutative curve.

So, let us look at a real-life example. Once again, take the Kleinian quartic In a previous course-post we recalled that there is an action by automorphisms on the Klein quartic K by the finite simple group of order 168. Hence, we can form the noncommutative Klein-quartic (take affine pieces consisting of complements of orbits and do the skew-group algebra construction on them and then glue these pieces together again).

We have also seen that the orbits are classified under a Belyi-map and that this map had the property that over any point of there is an orbit consisting of 168 points whereas over 0 (resp. 1 and ) there is an orbit consisting of 56 (resp. 84 and 24 points).

So what is the noncommutative manifold associated to the noncommutative Kleinian? Well, it looks like the picture we had at the start of this post For all but three points of the Riemann sphere we have one point and one loop (corresponding to a simple 168-dimensional representation of ) together with clusters of infinitesimally nearby points lying over 0,1 and (the cluster over 0 is depicted, the two others only indicated).

Over 0 we have three points connected by the diagram

where each of the vertices corresponds to a simple 56-dimensional representation. Over 1 we have a cluster of two points corresponding to 84-dimensional simples and connected by the picture we had in the example).

Finally, over we have the most interesting cluster, consisting of the seven dwarfs (each corresponding to a simple representation of dimension 24) and connected to each other via the picture

Again, this noncommutative manifold gives us all information needed to give a complete classification of all finite dimensional -representations. One can prove that all exceptional clusters of points for a noncommutative curve are connected by a cyclic quiver as the ones above. However, these examples are still pretty tame (in more than one sense) as these noncommutative algebras are finite over their centers, are Noetherian etc. The situation will become a lot wilder when we come to exotic situations such as the noncommutative manifold of …