Category: Mathematics

Back when I was preparing my book on the large sieve, around 2007 (the almost famous one), one question I struggled with was to find an upper bound for the largest dimension — say — of an irreducible complex representation of a finite group of Lie type , such as as or (in fact, especially the latter, since the first application I had in mind involved this group, the symplectic similitudes over a finite field.) Here, in my mind, the main variable was the size of the finite field, and not the parameter or (i.e., not the rank of the underlying group of Lie type).

I was pretty much convinced that this must be a well-known question with an easy answer (it the sense that anything that is known is “easy”, at least if one only needs to cite it to use it…) However, despite much Google search and even more Mathscinet search, I didn’t find this. I ended up asking an expert on the theory of finite groups of Lie type, who also had no reference in mind, but who could sketch how the bound I was expecting should follow from the theory of Deligne-Lusztig of (generalized) characters of finite groups of Lie type. This bound is of the type

for a (connected) semisimple group of dimension and rank , when grows. The underlying heuristic is quite simple: among the representations of , there are the “principal series”, parametrized by the characters of a maximal torus, and there are about of them, each of the same dimension . But on the other hand one knows that

(a basic fact of representation theory of finite groups), so we get something like

(the last equivalence is quite easy in concrete cases, and natural from general principles about the number of solutions of equations over finite fields), and if one believes that should be equal or close to the maximum, the guess above follows.

Incidentally, I think I already knew that a reverse inequality

is always true (in the generality above) because there is an explicit representation of that dimension, the Steinberg representation.

“At any rate”, I succeeded in getting a proof of a bound of the type

with an (explicit) implied constant depending only on the rank , learning the basic language of Deligne-Lusztig characters in doing so. (In fact, for or with large enough with respect to or , one gets an exact formula which immediately implies this.)

Now a few weeks ago, I saw a preprint of Larsen, Malle and Tiep appear on arXiv. The title The largest irreducible representations of simple groups is this time explicit enough, and the paper contains beautiful results, targeted towards the case of large rank in the case of groups of Lie type, because… the case of large was treated in a paper of G. M. Seitz in 1989!

However, the title of the latter paper is Cross-characteristic embeddings of finite groups of Lie type; the relevant section 2 is entitled An upper bound for the degrees of irreducible modules in this paper; and finally the Math Review (by R.W. Carter), although it does mention this section as being of independent interest, states

Then if is an absolutely irreducible -module of arbitrary characteristic we have , where is a maximal torus of of minimal order

This is perfectly limpid, but note that it fails to mention any of the keywords upper bound, maximal dimension or representation that I had been using in my frantic searches before…

I don’t know what is the moral of the story here. Knowing about this paper of Seitz (e.g., by asking around more systematically, or asking online, etc) would have saved me a fair amount of time, indeed. But then I wouldn’t have looked further into the theory of Deligne-Lusztig characters, which is very beautiful. To understand this theory, I also wouldn’t have had to look more deeply, and more concretely, into the structure theory of finite groups of Lie type and of algebraic groups in general. I would therefore know much less today, even though I would have known much earlier (and in greater generality!) this exact statement… But if I had known about Seitz’s work earlier, I would have been able to properly acknowledge this priority, of course — which I am of course happy to do here (whether there will be another printing of my book to update it is far from clear…)

In other words, beware which questions you ask: sometimes the worst thing that could happen is to get a precise answer…

Here’s a last remark: Seitz’s proof is somewhat like the one I had constructed, and relies on Deligne-Lusztig characters, and further work of Lusztig (I avoided some of the refinements of the latter by sticking with groups with connected centers which, for reasons amusingly related with the Langlands duality of reductive groups, are the “good” groups of Lie type for such arguments…) As Seitz observes, it would be interesting to have an elementary proof of these facts.

(End note: it is well-known that the corresponding question about lower bounds on the smallest dimension of a non-trivial representation of a finite group is of considerable importance in many applications of harmonic analysis on finite groups; for the simplest case of , the lower bound is , and this can be proved elementarily, as O. Dinai recalled me and others during last Friday’s train ride from Lausanne…)

(Update, 4.2.2011: I have updated, on my page of minor writings, the note containing the self-contained extract of my book with its proof of the upper-bound to mention Seitz’s priori result.)

Somewhere, there must be a regulation that says that the volume of beers must be indicated very precisely in Europe (or only in Switzerland?) to avoid any ambiguity:

On the other hand, this excellent idea does not seem to have been implemented yet (click for a better view):

The special semester on Group Actions in Number Theory, organized by Ph. Michel and myself, started last week with our Winter School.

We were lucky to have J.-P. Serre give a short course on equidistribution, with emphasis on questions related to the Sato-Tate conjecture and its variants. Here are a few things I’ve learnt (from the lectures and discussions afterwards):

(1) Bourbaki writes

for the unit circle, the projective plane and the affine 3-space respectively, because only for the third it is true that

(2) Contrary to popular (e.g., mine…) belief, there is one canonical finite field besides the fields of prime order. It is , which is canonical because there is a unique irreducible quadratic polynomial over , so that

(3) For the same reason, Bourbaki regrets the notation

for all finite fields, because it was their tradition to use bold fonts exclusively for objects which are completely canonical. Serre gave an example of a statement in his paper on Propriétés galoisiennes des points de torsion des courbes elliptiques which, if read too quickly, could give the impression of leading to a contradiction or a mistake — but only if one believes that is an unambiguous notation…

And finally — any links to mathematical brilliance will be left for the reader to contemplate — I learnt that around the 1940’s in Southern France, wine was the usual drink in middle and high-school lunches and dinners.

The title is a pretty facile pun, I admit, but I probably can’t manage much wittier after coming back from the AMS-MAA Joint Mathematical Meeting through a four hour delay in Memphis snow and another hour in Amsterdam exchanging an apparently broken 737 for a sounder one.

I had been invited by A. Salehi Golsefidy and A. Lubotzky to participate in their special session on expander graphs (slides to be found here), and since I had never attended such a big meeting, the occasion seemed as good as it could get — other tempting events happening then included Lubotzky’s Colloquium Lectures on expanders and the invited addresses of K. Soundararajan and A. Venkatesh.

Jordan Ellenberg also gave a talk at the expander session on our joint work with C. Hall, already discussed previously. This meant that I could rely on his gastronomical acumen for my first meals in New Orleans — a topic not to be tossed aside lightly (nor thrown away with great force, either). His heuristic technique was perfectly successful, and besides a fair amount of charcuterie, our dinner at Cochon included quite tasty fried alligator.

Random thoughts follow:

• Lubotzky’s lecture notes are excellent; especially fascinating is the mention that the first sighting of expanders, earlier typically attributed to Pinsker in 1973, has been found in a paper of Barzdin and Kolmogorov (Problemy Kibernetiki 1967, 19, 261–268), adding one more item to the long list of discoveries connected with Kolmogorov’s name. I hope to write a more complete post on this paper, which is very interesting (there is an English translation in Kolmogorov’s selected works).
• A mathematical meeting is even better when there is a good second-hand bookstore nearby, in this case, “Crescent City Books”; readers looking for a particular volume of the translations of the Proceedings of the Steklov Institute have a good chance of finding it there. For my own part, I bought a used copy of C. Spurgeon’s book on Shakespeare’s imagery, which is somewhat bardolatric, but fascinating anyway. (Quick do-you-think-like-Shakespeare-quizz: what is the first, or most vivid, image that comes to your mind about snails?)
• My last meal in New Orleans was the least satisfactory; however, a better one would probably not have left me enough time to wander around and stumble upon the excellent Idea Factory store, from which I left with some nice three-dimensional animal puzzles for my family…
• Coincidentally, my advisor, H. Iwaniec, received the Steele Prize for Mathematical Exposition during the meeting, especially for his graduate textbooks on automorphic forms — as I was one among many who learnt a lot from them, this was a great occasion to celebrate!
• Ancient astronomy was a bit on my mind, as I spent a lot of my time waiting in- or out-side airplanes listening to P. Glass’s opera Kepler, and reading a recent lively biography of Galileo. If you are doing the same, you may be interested in a new website of digitized books of old astronomy; I am planning to have a long look at some of the books where friends and foes of Galileo traded insults, insights, theoretical absurdities and experimental masterpieces…