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Tag: representations

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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Galois’ last letter

“Ne pleure pas, Alfred ! J’ai besoin de tout mon courage pour mourir à vingt ans!”

We all remember the last words of Evariste Galois to his brother Alfred. Lesser known are the mathematical results contained in his last letter, written to his friend Auguste Chevalier, on the eve of his fatal duel. Here the final sentences :



Tu prieras publiquement Jacobi ou Gauss de donner leur avis non sur la verite, mais sur l’importance des theoremes.
Apres cela il se trouvera, j’espere, des gens qui trouvent leur profis a dechiffrer tout ce gachis.
Je t’embrasse avec effusion.
E. Galois, le 29 Mai 1832

A major result contained in this letter concerns the groups L2(p)=PSL2(Fp), that is the group of 2×2 matrices with determinant equal to one over the finite field Fp modulo its center. L2(p) is known to be simple whenever p5. Galois writes that L2(p) cannot have a non-trivial permutation representation on fewer than p+1 symbols whenever p>11 and indicates the transitive permutation representation on exactly p symbols in the three ‘exceptional’ cases p=5,7,11.

Let α=[1101] and consider for p=5,7,11 the involutions on PFp1=Fp (on which L2(p) acts via Moebius transformations)

π5=(0,)(1,4)(2,3)π7=(0,)(1,3)(2,6)(4,5)π11=(0,)(1,6)(3,7)(9,10)(5,8)(4,2)

(in fact, Galois uses the involution  (0,)(1,2)(3,6)(4,8)(5,10)(9,7) for p=11), then L2(p) leaves invariant the set consisting of the p involutions Π=αiπpαi : 1ip. After mentioning these involutions Galois merely writes :

Ainsi pour le cas de p=5,7,11, l’equation modulaire s’abaisse au degre p.
En toute rigueur, cette reduction n’est pas possible dans les cas plus eleves.

Alternatively, one can deduce these permutation representation representations from group isomorphisms. As L2(5)A5, the alternating group on 5 symbols, L2(5) clearly acts transitively on 5 symbols.

Similarly, for p=7 we have L2(7)L3(2) and so the group acts as automorphisms on the projective plane over the field on two elements PF22 aka the Fano plane, as depicted on the left.

This finite projective plane has 7 points and 7 lines and L3(2) acts transitively on them.

For p=11 the geometrical object is a bit more involved. The set of non-squares in F11 is

1,3,4,5,9

and if we translate this set using the additive structure in F11 one obtains the following 11 five-element sets

1,3,4,5,9,2,4,5,6,10,3,5,6,7,11,1,4,6,7,8,2,5,7,8,9,3,6,8,9,10,

4,7,9,10,11,1,5,8,10,11,1,2,6,9,11,1,2,3,7,10,2,3,4,8,11

and if we regard these sets as ‘lines’ we see that two distinct lines intersect in exactly 2 points and that any two distinct points lie on exactly two ‘lines’. That is, intersection sets up a bijection between the 55-element set of all pairs of distinct points and the 55-element set of all pairs of distinct ‘lines’. This is called the biplane geometry.

The subgroup of S11 (acting on the eleven elements of F11) stabilizing this set of 11 5-element sets is precisely the group L2(11) giving the permutation representation on 11 objects.

An alternative statement of Galois’ result is that for p>11 there is no subgroup of L2(p) complementary to the cyclic subgroup

Cp=[1x01] : xFp

That is, there is no subgroup such that set-theoretically L2(p)=F×Cp (note this is of courese not a group-product, all it says is that any element can be written as g=f.c with fF,cCp.

However, in the three exceptional cases we do have complementary subgroups. In fact, set-theoretically we have

L2(5)=A4×C5L2(7)=S4×C7L2(11)=A5×C11

and it is a truly amazing fact that the three groups appearing are precisely the three Platonic groups!

Recall that here are 5 Platonic (or Scottish) solids coming in three sorts when it comes to rotation-automorphism groups : the tetrahedron (group A4), the cube and octahedron (group S4) and the dodecahedron and icosahedron (group A5). The “4” in the cube are the four body diagonals and the “5” in the dodecahedron are the five inscribed cubes.

That is, our three ‘exceptional’ Galois-groups correspond to the three Platonic groups, which in turn correspond to the three exceptional Lie algebras E6,E7,E8 via McKay correspondence (wrt. their 2-fold covers). Maybe I’ll detail this latter connection another time. It sure seems that surprises often come in triples…

Finally, it is well known that L2(5)A5 is the automorphism group of the icosahedron (or dodecahedron) and that L2(7) is the automorphism group of the Klein quartic.

So, one might ask : is there also a nice curve connected with the third group L2(11)? Rumour has it that this is indeed the case and that the curve in question has genus 70… (to be continued).

Reference

Bertram Kostant, “The graph of the truncated icosahedron and the last letter of Galois”

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the McKay-Thompson series

Monstrous moonshine was born (sometime in 1978) the moment John McKay realized that the linear term in the j-function

j(q)=1q+744+196884q+21493760q2+864229970q3+

is surprisingly close to the dimension of the smallest non-trivial irreducible representation of the monster group, which is 196883. Note that at that time, the Monster hasn’t been constructed yet, and, the only traces of its possible existence were kept as semi-secret information in a huge ledger (costing 80 pounds…) kept in the Atlas-office at Cambridge. Included were 8 huge pages describing the character table of the monster, the top left fragment, describing the lower dimensional irreducibles and their characters at small order elements, reproduced below

If you look at the dimensions of the smallest irreducible representations (the first column) : 196883, 21296876, 842609326, … you will see that the first, second and third of them are extremely close to the linear, quadratic and cubic coefficient of the j-function. In fact, more is true : one can obtain these actual j-coefficients as simple linear combination of the dimensions of the irrducibles :

{196884=1+19688321493760=1+196883+21296876864229970=2×1+2×196883+21296876+842609326

Often, only the first relation is attributed to McKay, whereas the second and third were supposedly discovered by John Thompson after MKay showed him the first. Marcus du Sautoy tells a somewhat different sory in Finding Moonshine :

McKay has also gone on to find these extra equations, but is was Thompson who first published them. McKay admits that “I was a bit peeved really, I don’t think Thompson quite knew how much I knew.”

By the work of Richard Borcherds we now know the (partial according to some) explanation behind these numerical facts : there is a graded representation V=iVi of the Monster-group (actually, it has a lot of extra structure such as being a vertex algebra) such that the dimension of the i-th factor Vi equals the coefficient f qi in the j-function. The homogeneous components Vi being finite dimensional representations of the monster, they decompose into the 194 irreducibles Xj. For the first three components we have the decompositions

{V1=X1X2V2=X1X2X3V3=X12X22X3X4

Calculating the dimensions on both sides give the above equations. However, being isomorphisms of monster-representations we are not restricted to just computing the dimensions. We might as well compute the character of any monster-element on both sides (observe that the dimension is just the character of the identity element). Characters are the traces of the matrices describing the action of a monster-element on the representation and these numbers fill the different columns of the character-table above.

Hence, the same integral combinations of the character values of any monster-element give another q-series and these are called the McKay-Thompson series. John Conway discovered them to be classical modular functions known as Hauptmoduln.

In most papers and online material on this only the first few coefficients of these series are documented, which may be just too little information to make new discoveries!

Fortunately, David Madore has compiled the first 3200 coefficients of all the 172 monster-series which are available in a huge 8Mb file. And, if you really need to have more coefficients, you can always use and modify his moonshine python program.

In order to reduce bandwidth, here a list containing the first 100 coefficients of the j-function

jfunct=[196884, 21493760, 864299970, 20245856256, 333202640600, 4252023300096, 44656994071935, 401490886656000, 3176440229784420, 22567393309593600, 146211911499519294, 874313719685775360, 4872010111798142520, 25497827389410525184, 126142916465781843075, 593121772421445058560, 2662842413150775245160, 11459912788444786513920, 47438786801234168813250, 189449976248893390028800, 731811377318137519245696, 2740630712513624654929920, 9971041659937182693533820, 35307453186561427099877376, 121883284330422510433351500, 410789960190307909157638144, 1353563541518646878675077500, 4365689224858876634610401280, 13798375834642999925542288376, 42780782244213262567058227200, 130233693825770295128044873221, 389608006170995911894300098560, 1146329398900810637779611090240, 3319627709139267167263679606784, 9468166135702260431646263438600, 26614365825753796268872151875584, 73773169969725069760801792854360, 201768789947228738648580043776000, 544763881751616630123165410477688, 1452689254439362169794355429376000, 3827767751739363485065598331130120, 9970416600217443268739409968824320, 25683334706395406994774011866319670, 65452367731499268312170283695144960, 165078821568186174782496283155142200, 412189630805216773489544457234333696, 1019253515891576791938652011091437835, 2496774105950716692603315123199672320, 6060574415413720999542378222812650932, 14581598453215019997540391326153984000, 34782974253512490652111111930326416268, 82282309236048637946346570669250805760, 193075525467822574167329529658775261720, 449497224123337477155078537760754122752, 1038483010587949794068925153685932435825, 2381407585309922413499951812839633584128, 5421449889876564723000378957979772088000, 12255365475040820661535516233050165760000, 27513411092859486460692553086168714659374, 61354289505303613617069338272284858777600, 135925092428365503809701809166616289474168, 299210983800076883665074958854523331870720, 654553043491650303064385476041569995365270, 1423197635972716062310802114654243653681152, 3076095473477196763039615540128479523917200, 6610091773782871627445909215080641586954240, 14123583372861184908287080245891873213544410, 30010041497911129625894110839466234009518080, 63419842535335416307760114920603619461313664, 133312625293210235328551896736236879235481600, 278775024890624328476718493296348769305198947, 579989466306862709777897124287027028934656000, 1200647685924154079965706763561795395948173320, 2473342981183106509136265613239678864092991488, 5070711930898997080570078906280842196519646750, 10346906640850426356226316839259822574115946496, 21015945810275143250691058902482079910086459520, 42493520024686459968969327541404178941239869440, 85539981818424975894053769448098796349808643878, 171444843023856632323050507966626554304633241600, 342155525555189176731983869123583942011978493364, 679986843667214052171954098018582522609944965120, 1345823847068981684952596216882155845897900827370, 2652886321384703560252232129659440092172381585408, 5208621342520253933693153488396012720448385783600, 10186635497140956830216811207229975611480797601792, 19845946857715387241695878080425504863628738882125, 38518943830283497365369391336243138882250145792000, 74484518929289017811719989832768142076931259410120, 143507172467283453885515222342782991192353207603200, 275501042616789153749080617893836796951133929783496, 527036058053281764188089220041629201191975505756160, 1004730453440939042843898965365412981690307145827840, 1908864098321310302488604739098618405938938477379584, 3614432179304462681879676809120464684975130836205250, 6821306832689380776546629825653465084003418476904448, 12831568450930566237049157191017104861217433634289960, 24060143444937604997591586090380473418086401696839680, 44972195698011806740150818275177754986409472910549646, 83798831110707476912751950384757452703801918339072000]

This information will come in handy when we will organize our Monstrous Easter Egg Race, starting tomorrow at 6 am (GMT)…

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