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Tag: Grothendieck

Klein’s dessins d’enfant and the buckyball

We saw that the icosahedron can be constructed from the alternating group A5 by considering the elements of a conjugacy class of order 5 elements as the vertices and edges between two vertices if their product is still in the conjugacy class.

This description is so nice that one would like to have a similar construction for the buckyball. But, the buckyball has 60 vertices, so they surely cannot correspond to the elements of a conjugacy class of A5. But, perhaps there is a larger group, somewhat naturally containing A5, having a conjugacy class of 60 elements?

This is precisely the statement contained in Galois’ last letter. He showed that 11 is the largest prime p such that the group L2(p)=PSL2(Fp) has a (transitive) permutation presentation on p elements. For, p=11 the group L2(11) is of order 660, so it permuting 11 elements means that this set must be of the form X=L2(11)/A with AL2(11) a subgroup of 60 elements… and it turns out that AA5

Actually there are TWO conjugacy classes of subgroups isomorphic to A5 in L2(11) and we have already seen one description of these using the biplane geometry (one class is the stabilizer subgroup of a ‘line’, the other the stabilizer subgroup of a point).

Here, we will give yet another description of these two classes of A5 in L2(11), showing among other things that the theory of dessins d’enfant predates Grothendieck by 100 years.

In the very same paper containing the first depiction of the Dedekind tessellation, Klein found that there should be a degree 11 cover PC1PC1 with monodromy group L2(11), ramified only in the three points 0,1, such that there is just one point lying over , seven over 1 of which four points where two sheets come together and finally 5 points lying over 0 of which three where three sheets come together. In 1879 he wanted to determine this cover explicitly in the paper “Ueber die Transformationen elfter Ordnung der elliptischen Funktionen” (Math. Annalen) by describing all Riemann surfaces with this ramification data and pick out those with the correct monodromy group.




He manages to do so by associating to all these covers their ‘dessins d’enfants’ (which he calls Linienzuges), that is the pre-image of the interval [0,1] in which he marks the preimages of 0 by a bullet and those of 1 by a +, such as in the innermost darker graph on the right above. He even has these two wonderful pictures explaining how the dessin determines how the 11 sheets fit together. (More examples of dessins and the correspondences of sheets were drawn in the 1878 paper.)

The ramification data translates to the following statements about the Linienzuge : (a) it must be a tree ( has one preimage), (b) there are exactly 11 (half)edges (the degree of the cover),
(c) there are 7 +-vertices and 5 o-vertices (preimages of 0 and 1) and (d) there are 3 trivalent o-vertices and 4 bivalent +-vertices (the sheet-information).

Klein finds that there are exactly 10 such dessins and lists them in his Fig. 2 (left). Then, he claims that one the two dessins of type I give the correct monodromy group. Recall that the monodromy group is found by giving each of the half-edges a number from 1 to 11 and looking at the permutation τ of order two pairing the half-edges adjacent to a +-vertex and the order three permutation σ listing the half-edges by cycling counter-clockwise around a o-vertex. The monodromy group is the group generated by these two elements.

Fpr example, if we label the type V-dessin by the numbers of the white regions bordering the half-edges (as in the picture Fig. 3 on the right above) we get
σ=(7,10,9)(5,11,6)(1,4,2) and τ=(8,9)(7,11)(1,5)(3,4).

Nowadays, it is a matter of a few seconds to determine the monodromy group using GAP and we verify that this group is A11.

Of course, Klein didn’t have GAP at his disposal, so he had to rule out all these cases by hand.

gap> g:=Group((7,10,9)(5,11,6)(1,4,2),(8,9)(7,11)(1,5)(3,4));
Group([ (1,4,2)(5,11,6)(7,10,9), (1,5)(3,4)(7,11)(8,9) ])
gap> Size(g);
19958400
gap> IsSimpleGroup(g);
true

Klein used the fact that L2(11) only has elements of orders 1,2,3,5,6 and 11. So, in each of the remaining cases he had to find an element of a different order. For example, in type V he verified that the element τ.(σ.τ)3 is equal to the permutation (1,8)(2,10,11,9,6,4,5)(3,7) and consequently is of order 14.

Perhaps Klein knew this but GAP tells us that the monodromy group of all the remaining 8 cases is isomorphic to the alternating group A11 and in the two type I cases is indeed L2(11). Anyway, the two dessins of type I correspond to the two conjugacy classes of subgroups A5 in the group L2(11).

But, back to the buckyball! The upshot of all this is that we have the group L2(11) containing two classes of subgroups isomorphic to A5 and the larger group L2(11) does indeed have two conjugacy classes of order 11 elements containing exactly 60 elements (compare this to the two conjugacy classes of order 5 elements in A5 in the icosahedral construction). Can we construct the buckyball out of such a conjugacy class?

To start, we can identify the 12 pentagons of the buckyball from a conjugacy class C of order 11 elements. If xC, then so do x3,x4,x5 and x9, whereas the powers x2,x6,x7,x8,x10 belong to the other conjugacy class. Hence, we can divide our 60 elements in 12 subsets of 5 elements and taking an element x in each of these, the vertices of a pentagon correspond (in order) to  (x,x3,x9,x5,x4).

Group-theoretically this follows from the fact that the factorgroup of the normalizer of x modulo the centralizer of x is cyclic of order 5 and this group acts naturally on the conjugacy class of x with orbits of size 5.

Finding out how these pentagons fit together using hexagons is a lot subtler… and in The graph of the truncated icosahedron and the last letter of Galois Bertram Kostant shows how to do this.



Fix a subgroup isomorphic to A5 and let D be the set of all its order 2 elements (recall that they form a full conjugacy class in this A5 and that there are precisely 15 of them). Now, the startling observation made by Kostant is that for our order 11 element x in C there is a unique element aD such that the commutator b=[x,a]=x1a1xa belongs again to D. The unique hexagonal side having vertex x connects it to the element b.xwhich belongs again to C as b.x=(ax)1.x.(ax).

Concluding, if C is a conjugacy class of order 11 elements in L2(11), then its 60 elements can be viewed as corresponding to the vertices of the buckyball. Any element xC is connected by two pentagonal sides to the elements x3 and x4 and one hexagonal side connecting it to τx=b.x.

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The F_un folklore

All esoteric subjects have their own secret (sacred) texts. If you opened the Da Vinci Code (or even better, the original The Holy blood and the Holy grail) you will known about a mysterious collection of documents, known as the “Dossiers secrets“, deposited in the Bibliothèque nationale de France on 27 April 1967, which is rumoured to contain the mysteries of the Priory of Sion, a secret society founded in the middle ages and still active today…

The followers of F-un, for F1 the field of one element, have their own collection of semi-secret texts, surrounded by whispers, of which they try to decode every single line in search of enlightenment. Fortunately, you do not have to search the shelves of the Bibliotheque National in Paris, but the depths of the internet to find them as huge, bandwidth-unfriendly, scanned documents.

The first are the lecture notes “Lectures on zeta functions and motives” by Yuri I. Manin of a course given in 1991.

One can download a scanned version of the paper from the homepage of Katia Consani as a huge 23.1 Mb file. Of F-un relevance is the first section “Absolute Motives?” in which

“…we describe a highly speculative picture of analogies between arithmetics over Fq and over Z, cast in the language reminiscent of Grothendieck’s motives. We postulate the existence of a category with tensor product × whose objects correspond not only to the divisors of the Hasse-Weil zeta functions of schemes over Z, but also to Kurokawa’s tensor divisors. This neatly leads to teh introduction of an “absolute Tate motive” T, whose zeta function is s12π, and whose zeroth power is “the absolute point” which is teh base for Kurokawa’s direct products. We add some speculations about the role of T in the “algebraic geometry over a one-element field”, and in clarifying the structure of the gamma factors at infinity.” (loc.cit. p 1-2)

I’d welcome links to material explaining this section to people knowing no motives.

The second one is the unpublished paper “Cohomology determinants and reciprocity laws : number field case” by Mikhail Kapranov and A. Smirnov.

This paper features in blog-posts at the Arcadian Functor, in John Baez’ Weekly Finds and in yesterday’s post at Noncommutative Geometry.

You can download every single page (of 15) as a separate file from here. But, in order to help spreading the Fun-gospel, I’ve made these scans into a single PDF-file which you can download as a 2.6 Mb PDF. In the introduction they say :

“First of all, it is an old idea to interpret combinatorics of finite sets as the q1 limit of linear algebra over the finite field Fq. This had lead to frequent consideration of the folklore object F1, the “field with one element”, whose vector spaces are just sets. One can postulate, of course, that spec(F1) is the absolute point, but the real problem is to develop non-trivial consequences of this point of view.”

They manage to deduce higher reciprocity laws in class field theory within the theory of F1 and its field extensions F1n. But first, let us explain how they define linear algebra over these absolute fields.

Here is a first principle : in doing linear algebra over these fields, there is no additive structure but only scalar multiplication by field elements. So, what are vector spaces over the field with one element? Well, as scalar multiplication with 1 is just the identity map, we have that a vector space is just a set. Linear maps are just set-maps and in particular, a linear isomorphism of a vector space onto itself is a permutation of the set. That is, linear algebra over F1 is the same as combinatorics of (finite) sets.

A vector space over F1 is just a set; the dimension of such a vector space is the cardinality of the set. The general linear group GLn(F1) is the symmetric group Sn, the identification via permutation matrices (having exactly one 1 in every row and column)

Some people prefer to view an F1 vector space as a pointed set, the special element being the ‘origin’ 0 but as F1 doesnt have a zero, there is also no zero-vector. Still, in later applications (such as defining exact sequences and quotient spaces) it is helpful to have an origin. So, let us denote for any set S by S=S0. Clearly, linear maps between such ‘extended’ spaces must be maps of pointed sets, that is, sending 00.

The field with one element F1 has a field extension of degree n for any natural number n which we denote by F1n and using the above notation we will define this field as :

F1n=μn with μn the group of all n-th roots of unity. Note that if we choose a primitive n-th root ϵn, then μnCn is the cyclic group of order n.

Now what is a vector space over F1n? Recall that we only demand units of the field to act by scalar multiplication, so each ‘vector’ v determines an n-set of linear dependent vectors ϵniv. In other words, any F1n-vector space is of the form V with V a set of which the group μn acts freely. Hence, V has N=d.n elements and there are exactly d orbits for the action of μn by scalar multiplication. We call d the dimension of the vectorspace and a basis consists in choosing one representant for every orbits. That is,  B=b1,,bd is a basis if (and only if) V=ϵnjbi : 1id,1jn.

So, vectorspaces are free μn-sets and hence linear maps VW is a μn-map VW. In particular, a linear isomorphism of V, that is an element of GLd(F1n) is a μn bijection sending any basis element biϵnj(i)bσ(i) for a permutation σSd.

An F1n-vectorspace V is a free μn-set V of N=n.d elements. The dimension dimF1n(V)=d and the general linear group GLd(F1n) is the wreath product of Sd with μn×d, the identification as matrices with exactly one non-zero entry (being an n-th root of unity) in every row and every column.

This may appear as a rather sterile theory, so let us give an extremely important example, which will lead us to our second principle for developing absolute linear algebra.

Let q=pk be a prime power and let Fq be the finite field with q elements. Assume that q1 mod(n). It is well known that the group of units Fq is cyclic of order q1 so by the assumption we can identify μn with a subgroup of Fq.

Then, Fq=(Fq) is an F1n-vectorspace of dimension d=q1n. In other words, Fq is an F1n-algebra. But then, any ordinary Fq-vectorspace of dimension e becomes (via restriction of scalars) an F1n-vector space of dimension e(q1)n.

Next time we will introduce more linear algebra definitions (including determinants, exact sequences, direct sums and tensor products) in the realm the absolute fields F1n and remarkt that we have to alter the known definitions as we can only use the scalar-multiplication. To guide us, we have the second principle : all traditional results of linear algebra over Fq must be recovered from the new definitions under the vector-space identification Fq=(Fq)=F1n when n=q1. (to be continued)

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un-doing the Grothendieck?

(via the Arcadian Functor) At the time of the doing the Perelman-post someone rightfully commented that “making a voluntary retreat from the math circuit to preserve one’s own well-being (either mental, physical, scientific …)” should rather be called doing the Grothendieck as he was the first to pull this stunt.

On Facebook a couple of people have created the group The Petition for Alexander Grothendieck to Return from Exile. As you need to sign-up to Facebook to use this link and some of you may not be willing to do so, let me copy the description.

Alexander Grothendieck was born in Berlin, Germany on March 28, 1928. He was one of the most important and enigmatic mathematicians of the 20th century. After a lengthy and very productive career, highlighted by the awarding of the Fields Medal and the Crafoord Prize (the latter of which he declined), Grothendieck disappeared into the French countryside and ceased all mathematical activity. Grothendieck has lived in self-imposed exile since 1991.

We recently spotted Grothendieck in the “Gentleman’s Choice” bar in Montreal, Quebec. He was actually a really cool guy, and we spoke with him for quite some time. After a couple of rounds (on us) we were able to convince him to return from exile, under one stipulation – we created a facebook petition with 1729 mathematician members!

If 1729 mathematicians join this group, then Alexander Grothendieck will return from exile!!

1729 being of course the taxicab-curve number. The group posts convincing photographic evidence (see above) for their claim, has already 201 members (the last one being me) and has this breaking news-flash

Last week Grothendieck, or “the ‘Dieck” as we affectionately refer to him, returned to Montreal for a short visit to explain some of the theories he has been working on over the past decade. In particular, he explained how he has generalised the theory of schemes even further, to the extent that the Riemann Hypothesis and a Unified Field Theory are both trivial consequences of his work.

You know what to do!

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