Mathematics – Page 28 – E. Kowalski's blog

What’s special with commutators in the Weyl group of C5?

I have just added to my notes on representation theory the very cute formula of Frobenius that gives, in terms of irreducible characters, the number $N(g)$ of representations of a given element $g$ as a commutator $g=[x,y]=xyx^{-1}y^{-1}$ in a finite group $G$ :
$N(g)=|G|\sum_{\chi}\frac{\chi(g)}{\chi(1)},$
where $\chi$ runs over the irreducible (complex) characters of $G$ (this is Proposition 4.4.3 on page 118 of the last version of the notes).

I wanted to mention some applications, and had a vague memory that this was used to show that most or all elements in various simple groups are actual commutators. By searching around a bit, I found out easily that, indeed, there was a conjecture of Ore from 1951 to the effect that the set of commutators is equal to $G$ for any non-abelian finite simple group $G$ , and that (after various earlier works) this has recently been proved by Liebeck, O’Brien, Shalev and Tiep.

I mentioned this of course, but then I also wanted to give some example of non-commutator, and decided to look for this using Magma (the fact that I am recovering from a dental operation played a role in inciting me to find something distracting to do). Here’s what I found out.

First, a natural place to look for interesting examples is the class of perfect groups, of course not simple. This is also easy enough to implement since Magma has a database of perfect groups of “small” order. Either by brute force enumeration of all commutators or by implementing the Frobenius formula, I got the first case of a perfect group $G$ , of order $960$ , which contains only $840$ distinct commutators.

Then I wanted to know “what” this group really was. Magma gave it to me as a permutation group acting on $16$ letters, with an explicit set of $6$ generators, and with a list of $21$ relations, which was not very enlightening. However, looking at a composition series, it emerged that $G$ fits in an exact sequence
$1\rightarrow (\mathbf{Z}/2\mathbf{Z})^4\rightarrow G\rightarrow A_5\rightarrow 1.$
This was much better, since after a while it reminded me of one of my favorite types of groups: the Weyl groups $W_{g}$ of the symplectic groups $\mathrm{Sp}_{2g}$ (equivalently, the “generic” Galois group for the splitting field of a palindromic rational polynomial of degree $2g$ ), which fit in an relatively similar exact sequence
$1\rightarrow (\mathbf{Z}/2\mathbf{Z})^g\rightarrow W_g\rightarrow S_g\rightarrow 1.$
From there, one gets a strong suspicion that $G$ must be the commutator subgroup of $W_5$ , and this was easy to check (again with Magma, though this is certainly well-known; the drop of the rank of the kernel comes from looking at the determinant in the signed-permutation $5$ -dimensional representation, and the drop from $S_5$ to $A_5$ is of course from the signature.)

This identification is quite nice, obviously. In particular, it’s now possible to identify concretely which elements of $G$ are not commutators. It turns out that a single conjugacy class, of order $120$ , is the full set of missing elements. As a signed permutation matrix, it is the conjugacy class of
$g=\begin{pmatrix} 0& -1 & 0 & 0 & 0\\ 1& 0 & 0 & 0 & 0\\ 0& 0 & 0 & 1 & 0\\ 0& 0 & 1 & 0 & 0\\ 0& 0 & 0 & 0 & -1\end{pmatrix},$
and the reason it is not a commutator is that Magma tells us that all commutators in $G$ have trace in $\{-3,-2,0,1,2,5\}$ (always in the signed-permutation representation). Thus the trace $-1$ doesn’t fit…

At least, this is the numerical reason. I feel I should be able to give a theoretical explanation of this, but I haven’t succeeded for the moment. Part of the puzzlement is that this behavior seems to be special to $W_5$ , the Weyl group of the root system $C_5$ . Indeed, for $g\in\{2,3,4\}$ , the corresponding derived subgroup is not perfect, so the question does not arise (at least in the same way). And when $g\geq 6$ , the derived subgroup $G_g$ of $W_g$ is indeed perfect, but — experimentally! — it seems that all elements of $G_g$ are then commutators.

I haven’t found references to a study of this Ore-type question for those groups, so I don’t know if these “experimental” facts are in fact known to be true. Another question seems natural: does this special fact have any observable consequence, for instance in Galois theory? I don’t see how, but readers might have better insights…

(P.S. I presume that GAP or Sage would be equally capable of making the computations described here; I used Magma mostly because I know its language better.

P.P.S And the computer also tells us that even for the group $G$ above, all elements are the product of at most two commutators, which a commenter points out is also a simple consequence of the fact that there are more than $480$ commutators….

P.P.P.S To expand one of my own comments: the element $g$ above is a commutator in the group $W_5$ itself. For instance $g=[x,y]$ with
$x=\begin{pmatrix} 0& 0 & 0 & 0 & -1\\ 0& 1 & 0 & 0 & 0\\ 1& 0 & 0 & 0 & 0\\ 0& 0 & 1 & 0 & 0\\ 0& 0 & 0 & 1 & 0\end{pmatrix},$
and
$y=\begin{pmatrix} 1& 0 & 0 & 0 & 0\\ 0& 0 & 0 & 0 & -1\\ 0& 1 & 0 & 0 & 0\\ 0& 0 & 1 & 0 & 0\\ 0& 0 & 0 & -1 & 0\end{pmatrix},$
where $y\notin G$ .)

Fermat on Google

Today’s Google-Doodle…

Update on representation theory notes

After a long hiatus, I have just put up online an updated version of my lecture notes on representation theory. The delay was psychologically interesting: after a long period where I added material more or less in the order I wanted it to appear in the text, I started in June to proceed in much more chaotic (or random?) manner, with an explanation of the Larsen alternative for unitary groups coming before the Peter-Weyl theorem, and so son. Inly in the last few days did the text regain at least some coherence. (In particular, it took me a long time to finally sit up and write an account of the Peter-Weyl theorem that I felt to be at least somewhat motivated.)

There are still things missing before the notes contain all that I’d like, in particular at least a few pages of survey concerning the representation theory of some locally compact, non compact, groups. There are still a few weeks before the beginning of the new semester, however, and hopefully I will have time to do some work on this part in the coming weeks…

As usual, any remarks or corrections will be very appreciated!

Peeling onions with Peter and Weyl

I had promised a while ago to say more about the original proof of Peter-Weyl of the “completeness theorem” for compact groups: matrix coefficients of finite-dimensional irreducible representations span a dense subspace of $L^2(G)$ , for a compact group $G$ . With some delay, here we go…

Before giving just a brief overview of the strategy in the paper of Peter and Weyl, here are some mostly historical or psychological remarks that came to mind while looking at the paper and related sources:

If one had infinite amount of time to teach everything about representation theory, I guess most mathematicians would treat the Peter-Weyl theory fairly soon after finite groups, and before the representation theory of semisimple Lie groups; interestingly, the history was reversed: Weyl first developed the basic representation theory of the charmingly named “kontinuierlicher halb-einfacher Gruppen” in three papers in Mathematische Zeitschrift in 1925, before going to the proof of the completeness theorem;
Similarly, since expressing special functions using representations can be done with very concrete examples, and this “works” with the most elementary compact Lie groups, one might expect that this would have been done before the general theory; but history also proceeded in the opposite order: É. Cartan, who according to Vilenkin was the first to make the connection explicitly, writes very clearly in the very first lines of his paper “Sur la détermination d’un système orthogonal complet dans un espace de Riemann symétrique clos” (1928) that “Le présent mémoire a été inspiré par la lecture du beau mémoire où H. WEYL montre que les différentes représentations linéaires irréducibles d’un groupe continu clos fournissent un système orthogonal complet dans l’espace du groupe” [“the present memoir was inspired by reading the beautiful memoir where H. WEYL shows that the various irreducible linear representations of a continuous compact group give a complete orthogonal system in the space of the group”]; the paper which is referenced — note that the poor Peter is here also rather casually ignored, though the footnote gives the full reference — is the one of Peter and Weyl.
Peter and Weyl amusingly define a representation of a group $G$ as a map $E$ from $G$ to a space of matrices such that $E(st)=E(s)E(t)$ for all $s$ and $t$ in $G$ ; they do not ask that it be a homomorphism! However, as they observe, the matrix $E(1)$ is a projector, and the $E(s)$ preserve its image, so be restriction to the latter, one obtains a “standard” representation.
The paper is written in an interesting “intermediate style” of analysis. There are inequalities, estimates, limiting arguments, but no functional analysis. Although functions are integrated freely, there is no mention of measurability or integrability conditions (all functions are freely evaluated at the origin, also…). No function space is identified, no linear operator mentioned explicitly. Although some inequalities would today be immediately interpreted as the standard inequality $\|T(x)\|\leq \|T\|\|x\|$ for a continuous linear map $T$ acting on some normed vector space, there is no trace of such things.
The paper feels however quite modern in its formalism: a function on the group $G$ is thought of as being a “Gruppenzahl”, and denoted with a lower-case letter like $x$ or $z$ ; these are multiplied by convolution without a specific notation, etc. (I wonder if there could be here already an influence of quantum mechanics and the “q”-numbers that were infinite matrices?)

So what is the Peter-Weyl argument? I’ll first say how it differs from the modern treatments (at least, those I have seen): either for philosophical reasons (which is conceivable, in view of Weyl’s fairly constructivist ideas) or because abstract Hilbert space theory was not within their frame of thought, they do not use the type of argument that comes the most naturally to mind today: to show that the finite-dimensional matrix coefficients span a dense subspace, one shows that its orthogonal is zero. This reduces, for a given non-zero function $\varphi$ on $G$ , to finding a single finite-dimensional unitary representation $\rho$ for which $\varphi$ is not orthogonal to the corresponding space of matrix coefficients. Instead, Peter and Weyl more or less present an algorithmic way to, in principle, decompose $\varphi$ into a combination of matrix coefficients of finite-dimensional representations.

In both cases, however, the basic mechanism to produce the finite-dimensional representations is the same: one uses integral operators on $G$ , constructed as convolution operators using the “left” regular representation, and which therefore commute with the “right” regular one. Any non-zero eigenspace of such an operator is a finite-dimensional subrepresentation of the right-regular representation, and basically because any function gives a suitable integral operator, it is not too surprising that this gives enough finite-dimensional unitary representations.

This is the principle. The Peter-Weyl constructive method is maybe best illustrated in the case of the circle, where the theory becomes that of Fourier series (of course, the completeness in that case is older and has many different proofs, but in fact Weyl first implemented the strategy in the related case of Bohr’s almost periodic functions, to get the analogue of Parseval’s identity in that setting; this was in the same volume of Math. Annalen, in fact.) Here, given a 1-periodic function
$\varphi(t)=\sum_{h\in \mathbf{Z}}{c(h)e^{2i\pi h t}},$
what they do is basically to construct (by iterative procedures) the successive approximations
$\varphi_0,\quad \varphi_1,\quad,\ldots,\quad \varphi_k,\quad\ldots$
of the Fourier series such that
$\varphi_j=\sum_{h\in S_j}{c(h)e^{2i\pi h t}},$
where the sets of frequencies $S_j$ are increasing (for inclusion), and involve the successive Fourier coefficients with decreasing magnitude. In other words, $S_0$ is the set of frequencies where $|c(h)|$ is maximal, $S_1$ adds all frequencies for which $|c(h)|$ is the second-largest, etc. This is related to the previous discussion because the $c(h)$ are the eigenvalues of the convolution operator with kernel $\varphi$ , and $|c(h)|^2$ are those for $\varphi\star\check\varphi$ , the corresponding non-negative kernel.

Using a rather cute argument, Peter and Weyl are able to show that (the analogue for compact gropups of) this process leads to approximations which converge in $L^2$ to $\varphi$ , i.e., to a proof of the Parseval identity, which is the completeness theorem.

Altogether, for the circle, this proof might be a bit involved, but it is very nice and conceptual, and would certainly make interesting exercises in a functional analysis class. I had not seen it anywhere before, but it could be just personal ignorance of the literature… I also have no idea if the constructive aspect is actually numerically interesting.

To justify the title of the post: the way I think about this is that they “peel off” the largest contributions to the function $\varphi$ , one by one, as one may peel an onion…

Die kleinsche Flasche ist keine riemannsche Fläche

(Or in other words, the Klein bottle

is not a Riemann surface…)

I bought this little marvel in the shop of the excellent Giessen Mathematikum math museum. My two favorite exhibits were this playful shadow (a sculpture of Larry Kagan)

and a rather thought-provoking physical illustration of what is a “one in a million chance”