lieven le bruyn's blog
Posts tagged Borcherds
Dedekind or Klein ?
Apr 22nd
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)
the McKay-Thompson series
Mar 22nd
Monstrous moonshine was born (sometime in 1978) the moment John McKay realized that the linear term in the j-function
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 :
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 
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 
equals the coefficient f 
in the j-function. The homogeneous components being finite dimensional representations of the monster, they decompose into the 194 irreducibles 
. For the first three components we have the decompositions
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)…
thanks for linking
Jan 11th
google analytics
I’ve re-installed the Google analytics plugin on december 22nd, so it is harvesting data for three weeks only. Still, it is an interesting tool to gain insight in the social networking aspect of math-blogging, something I’m still very bad at…
Below the list of all blogs referring at least 10 times over this last three weeks. In brackets are the number of referrals and included are the average time Avg. they spend on this site, as well as the bounce back rate BB. It gives me the opportunity to link back to some of their posts, as a small token of gratitude. I may repeat this in the future, so please keep on linking…
Not Even Wrong (69) : Avg (1.05 min) BB (52.94%)
The most recent post of Peter is an update on the plagiarism scandal on the arXiv.
The n-category cafe (63) : Avg (2.13 min) BB (50%)
The one series I followed at the cafe lately was the Geometric Representation Theory course run by John Baez and James Dolan. They provide downloadable movies as well as notes.
Richard Borcherd’s blog (47) : Avg (1.53 min) BB (53.19%)
It is great to see that Borcherds has taken up blogging again, with a post on the uselessness of set theory.
The Arcadian functor (32) : Avg (3.45 min) BB (34.38 %)
It is clear from the low bounce-back rate and the high average time spend on this site, that Kea’s readers and mine have common interests. Often I feel that Kea and I are talking about the same topics, but that our language is so different, that it is difficult for me to spot the precise connection. I definitely should start (for myself) a translation-project of her M-theory posts.
RupertGee’s iBlog (23) : Avg (6.48 min) BB (34.7 %)
Surprisingly, and contrasting to my previous rant iTouch-people (or at least those coming here from Rupert Gee’s blog) sure take time to read the posts and look for more.
Ars Mathematica (22) : Avg (0:01 min) BB (77,2 %)
Well, the average time and bounce back rate say it all : people coming here from Ars Mathematica are not interested in longer posts…
iTouch Fans Forum (14) : Avg (2:07 min) BB (42.86 %)
Again, better statistics than I would have expected.
Vivatsgasse 7 (13) : Avg (1:51 min) BB (38.46 %)
I hope these guys haven’t completely given up on blogging as it is one of my favourites.
Sixth form mathematics (12) : Avg (1:40 min) BB (25 %)
My few old posts on LaTeXrender still draw referrals…
Strategic Boards (12) : Avg (0:01 min) BB (91.67 %)
People in strategic board games are not really in my game-posts it seems…
The Everything Seminar (11) : Avg (2:04 min) BB (72.73 %)
Greg Muller has been posting a couple of nice posts on chord diagrams, starting here.
Noncommutative Geometry (11) : Avg (3:36 min) BB (27.27 %)
Well, we are interested in the same thing viewed from different angles, so good average times and a low bounce back rate. Maybe, I should make another attempt to have cross-interaction between the two blogs.
symmetry and the monster
Jun 2nd
math-books
Mark Ronan has written a beautiful book intended for the general public on Symmetry and the Monster. The book’s main theme is the classification of the finite simple groups. It starts off with the introduction of groups by Galois, gives the classifivcation of the finite Lie groups, the Feit-Thompson theorem and the construction of several of the sporadic groups (including the Mathieu groups, the Fischer and Conway groups and clearly the (Baby)Monster), explains the Leech lattice and the Monstrous Moonshine conjectures and ends with Richard Borcherds proof of them using vertex operator algebras. As in the case of Music of the Primes it is (too) easy to be critical about notation. For example, whereas groups are just called symmetry groups, I don’t see the point of calling simple groups ‘atoms of symmetry’. But, unlike du Sautoy, Mark Ronan stays close to mathematical notation, lattices are just lattices, characer-tables are just that, j-function is what it is etc. And even when he simplifies established teminology, for example ‘cyclic arithmetic’ for modular arithmetic, ‘cross-section’ for involution centralizer, ‘mini j-functions’ for Hauptmoduln etc. there are footnotes (as well as a glossary) mentioning the genuine terms. Group theory is a topic with several colourful people including the three Johns John Leech, John McKay and John Conway and several of the historical accounts in the book are a good read. For example, I’ve never known that the three Conway groups were essentially discovered in just one afternoon and a few telephone exchanges between Thompson and Conway. This year I’ve tried to explain some of monstrous moonshine to an exceptionally good second year of undergraduates but failed miserably. Whereas I somehow managed to give the construction and proof of simplicity of Mathieu 24, elliptic and modular functions were way too difficult for them. Perhaps I’ll give it another (downkeyed) try using ‘Symmetry and the Monster’ as reading material. Let’s hope Oxford University Press will soon release a paperback (and cheaper) version.
capita selecta
Apr 22nd
Rather than going to the NOG
III Workshop I think it is more fun to give a talk for the Capita
Selecta-course for 2nd year students on “Monstrous Moonshine”. If
I manage to explain to them at least something, I think I am in good
shape for next year\’s Baby Geometry (first year) course. Besides,
afterwards I may decide to give some details of Borcherds\’ solution next year in my 3rd year
Geometry-course…(but this may just be a little bit
over-optimistic).
Anyway, this is what I plan to do in my
lecture : explain both sides of the McKay-observation
that
196 884 = 196 883 + 1that is, I\’ll give the action of the modular group on the upper-half plane and prove that its fundamental domain is just C using the modular j-function (left hand side) and sketch the importance of the Monster group and its representation theory (right hand side). Then I\’ll mention Ogg\’s observation that the only subgroups Gamma(0,p)+ of SL(2,Z) for which the fundamental domain has genus zero are the prime divisors p of teh order of the Monster and I\’ll come to moonshine conjecture of Conway and Norton (for those students who did hear my talk on Antwerp sprouts, yes both Conway and Simon Norton (via his SNORT-go) did appear there too…) and if time allows it, I\’ll sketch the main idea of the proof. Fortunately, Richard Borcherds has written some excellent expository papers I can use (see his papers-page and I also discovered a beautiful moonshine-page by Helena Verrill which will make my job a lot easier.
Btw. yesterday\’s Monster was taken from her other monster story…
Hi, I'm a professor of mathematics at the University of Antwerp, in Belgium. My research areas are non-commutative algebra and non-commutative geometry. Sometimes, you can also find me here :
