Page 37 – E. Kowalski's blog

This will only reinforce stereotypes…

How much wine should riot-control police be allowed at lunch time? Should they strike about it? Important questions…

The other theorem of Burnside

If a poll was held among mathematicians to identify “Burnside’s Theorem”, I guess that the result would be roughly evenly split between the irreducibility criterion for finite-dimensional representations, and the solvability of finite groups of order divisible by at most two distinct primes; far behind would come the so-called “Burnside lemma”, which I’ve heard is due to Frobenius or Cauchy, the partial results concerning Burnside’s problem, and presumably among stragglers and also-ran would be Schur’s Lemma and his work on hydrodynamics and complex function theory.

I’ve already discussed what head or tail I could make of Burnside’s proof of the irreducibility criterion, which is also incorporated in my representation theory notes (I’ve also put in there, in Section 2.7.4, a different proof based on Frobenius reciprocity which is quite cute, and presumably also well-known). I’ve just started discussing in class the $p^aq^b$ theorem. The account in the notes (Section 4.7.2) is a bit incomplete, as I’m struggling in my attempts to decide how to present in a motivated way how one comes to the integrality properties of characters which are the crucial tool (for a target audience with no prior exposure to algebraic integers). I found this easier to do on the board, and to imagine doing with hyperlinks, than within a book-like object!

I had not actually looked at the proof before the last few weeks. As far as I can see, and in striking contrast with the irreducibility criterion, Burnside’s proof is still the standard one (apart from terminological and notational changes; I do not count the purely group-theoretic arguments apparently found later by Thompson and others, which I know nothing about, save their existence). Part of the mystery of the statement is that, in fact, one proves something a bit different, and in fact weaker-looking: if $|G|=p^aq^b$ , then either $G$ is abelian, or it contains a proper normal subgroup. Then by induction on the order of a group $|G|$ (divisiblie by at most two primes), one concludes easily that in fact they are solvable.

In turn, the desired normal subgroup is constructed using a more general result, which I had never heard about, but which is certainly of independent interest, and has the virtue of being a general fact about all finite groups which illustrates some of the subtle ways in which irreducible (complex) representations and conjugacy classes try to be “dual”:

If $G$ is a finite group, $g\in G$ is a non-trivial element, $\rho$ a non-trivial irreducible complex representation of $G$ . If the dimension of $\rho$ is coprime to the order of the conjugacy class of $g$ , then either

The character value $\chi_{\rho}(g)$ is zero;

Or, the element $g$ is in the kernel of the composite projective representation
$\bar{\rho}\,:\, G\rightarrow \mathrm{GL}(E)\rightarrow \mathrm{PGL}(E)$
(where $E$ is the space of $\rho$ ).

The second part is the main point: if, for a given group $G$ , one can find a pair $(g,\rho)$ for which the result is applicable, and if the first part of the alternative can be excluded, then it follows that the kernel $N$ of $\bar{\rho}$ is a non-trivial normal subgroup. It could be that $N=G$ , but that is a very special case: then the image of $\rho$ is an abelian group, and either $\rho$ is an isomorphism with such an abelian group, or the kernel of $\rho$ itself is a proper normal subgroup.)

Where having $|G|=p^aq^b$ comes in, in applying this, is in the fact that it is not so easy in general to cook up integers dividing $|G|$ which are coprime to each other (these being the size $|g^{\sharp}|$ of the conjugacy class $g^{\sharp}$ of $g$ , and the dimension of the irreducible representation).

What can be done without much work is to find, for any $g\not=1$ , a representation of dimension not divisible by one prime (say $p\mid |G|$ ) for which $\chi_{\rho}(g)\not=0$ , using the orthogonality relations looked-at modulo $p$ ; and one can also find an element $g\not=1$ for which the conjugacy class has order coprime with another prime $q$ (looking at the partition of $G$ in conjugacy classes, modulo $q$ ). But there is no reason that this should ensure that $(\dim(\rho),|g^{\sharp}|)=1$ , except if $|G|$ is divisible only by those two primes! In that case, we see that $\dim(\rho)$ is a power of $p$ , and the conjugacy class of $g$ has size a power of $q$ , and everything works.

The proof of the result on characters above is a very convincing illustration of the usefulness of algebraic integers: one first shows by basic facts about them that there is always a divisibility
$\dim(\rho)\mid \chi_{\rho}(g) |g^{\sharp}|,$
in the ring of algebraic integers. Thus if, as we assumed, the dimension and the size of the conjugacy class are coprime, we get
$\dim(\rho)\mid \chi_{\rho}(g),$
and using the fact that the character value is a sum of $\dim(\rho)$ roots of unity, the conclusion is again not too hard from the properties of algebraic integers. (It is quite obvious when $\chi_{\rho}(g)\in\mathbf{Z}$ , of course, since one knows that
$|\chi_{\rho}(g)|\leq \dim\rho$ , and one must see that a similar argument applies in general…)

Amusingly, one can use the divisibility relation in the “opposite” direction: if the character value is non-zero and coprime (in the ring of algebraic integers) with the dimension of the representation, then $\dim(\rho)$ divides the size of the conjugacy class. I don’t know if there are applications of this, but this can be seen in practice, e.g., for $G=\mathrm{GL}_2(\mathbf{F}_p)$ when $\rho$ is a Steinberg representation of dimension $p$ and $g$ is a semisimple (split or non-split) conjugacy class, where the character value is a root of unity, hence coprime with $p$ , and indeed the conjugacy classes in question have order $p(p+1)$ or $p(p-1)$ (in the split and non-split case, respectively.)

The group algebra as “universal endomorphisms”

My lecture notes on representation theory now include some discussion of the group algebra (though I haven’t yet covered this in class). The motivation I am emphasizing (though the emphasis comes from the unorthodox practice of making it the topic of Exercise 3.1.5 in Chapter 3…) is that the elements of the group algebra $k(G)$ correspond exactly, through their action, to what might be called “universal” endomorphisms of representations (though “functorial” is the more proper term).

More precisely, a “universal endomorphism” is a collection $\varepsilon_{\rho}$ of linear maps
$\varepsilon_{\rho}\,:\, E\longrightarrow E$
defined for all representations
$\rho\,:\, G\longrightarrow \mathrm{GL}(E),$
in such a way that for any $G$ -homomorphism
$\Phi\,:\, E\longrightarrow F,$
we have
$\Phi\circ \varepsilon_{\rho}=\varepsilon_{\tau}\circ \Phi$
(which means that the obvious square diagram, which I am unable to reproduce with the LaTeX plugin of this blog, commutes). The elementary result of the exercise is that, for any such collection, there is a unique element $a\in k(G)$ such that
$\varepsilon_{\rho}(v)=\rho(a)v$
for all representations $\rho$ , and all vectors $v$ in the space of $\rho$ . (Hint: the element $a$ is simply $\varepsilon_{k(G)}(1)$ — where $k(G)$ has of course the left multiplication action –, and to show that it has the required property, one first deals with $\rho=k(G)$ itself using the $G$ -homomorphisms given by right-multiplication by some element $b$ , etc.)

The reason I like this viewpoint (besides the fact that it is not discussed this way in the books I know, and one likes to be at least infinitesimally original) is that it makes it clear that there must be some nice formula for the projectors on isotypic components for finite groups (in good characteristic): indeed, it is easy to check that, for a fixed irreducible representation $\pi$ , the family
$\varepsilon_{\rho}=(\text{projector on the }\pi-\text{isotypic component of }\rho)$
satisfies the functoriality property above. Thus this must be given by the action of some element of the group algebra, and by inspection, one is led to the “right” one (using the orthogonality of characters, in my notes, though maybe there is an even more intrinsic reason?) To me, this seems to demystify the result: it is not a question of checking a miraculous formula, but of using some reasonable enough reasoning to find the right one.

(Note: the isotypic projectors are also themselves $G$ -homomorphisms; it is of course not difficult to check that a “universal” endomorphism is a universal $G$ -homomorphism if and only if the corresponding element $a$ is in the center of the group algebra.)

Bubbles

To continue with the topic of liquids:

Esperantism complicates knots

One of the results in the new preprint of Gromov and Guth that I mentioned in my previous post is a rather striking application of expanders to a problem of knot theory: a new proof of the fact (only recently proved by Pardon) that there exist knots $K$ (in $\mathbf{R}^3$ ) with arbitrarily large distortion $d(K)$ , where the latter is defined by
$d(K)=\inf_{k}\sup_{x,y \in k} \frac{d_k(x,y)}{|x-y|},$
where $k$ runs over knots isotopic to $K$ , and the distance in $k$ in the numerator being the distance between points on the curve that “is” $k$ . In other words, the distortion is large when, whichever way we put the knot in space, there are points on it which are “physically close” in the ambient space, but far away if one is forced to connect them along the knot itself.

(Here and henceforth, any knot-theoretic blunders are obviously my own unenlightened ones…)

Gromov had asked whether the distortion is unbounded in the early 80’s, and Pardon gave the first examples (there is a blog post here that explains his work).

The examples of Gromov and Guth (Section 4 of their paper, which they describe as “the most interesting”…) arise from the following context: let $M$ be a fixed hyperbolic arithmetic 3-manifold; let $M_i\rightarrow M$ be any sequence of arithmetic coverings of $M$ with degree $d_i$ tending to infinity as $i$ does; then, by results of Hilden and Montesinos from the 1970’s, there are knots $K_i$ associated to $M_i$ by the property that $M_i$ can be written as a $3$ -cover of the sphere $\mathbf{S}^3$ ramified over $K_i$ . Then Gromov and Guth prove that
$d(K_i)\gg h_i V_i,$
where $h_i$ is the Cheeger constant for $M_i$ , and
$V_i=d_i\mathrm{Vol}(M)$
is the hyperbolic volume of $M_i$ .

Now, by the Cheeger-Buser principle, we have
$h_i\gg \lambda_1(M_i),$
where $\lambda_1(M_i)$ is the first positive eigenvalue of the hyperbolic Laplace operator on $M_i$ . By Property (τ) for $M$ — which uses the arithmetic property of the coverings –, this first eigenvalue is bounded away from zero, and hence the Gromov-Guth inequality gives
$\lim_{i\rightarrow +\infty} d(K_i)=+\infty.$

The link with expanders is the Brooks-Burger comparison principle: the $\lambda_1$ of the hyperbolic Laplace operator is (up to constants depending only on $M$ and a choice of generators of its fundamental group) of the same order of magnitude as the first positive eigenvalue for the Cayley-Schreier graphs $\Gamma_i$ associated to the covering $M_i\rightarrow M$ .

This type of setting may remind (very) faithful readers of the discussion of some of my own results with J. Ellenberg and C. Hall, where expanders turned out to be slightly overkill, a weaker degree of expansion of the graphs — which we called “esperantism” — being sufficient for our purposes.

I think — but I may have missed some problem! — that for the purpose of constructing knots with large distortion, the same basic idea also applies. Namely, if we take for $M_i$ some congruence coverings with degree $d_i\asymp i^k$ for some fixed $k>0$ , an inequality of Brooks shows that
$h_i\gg h(\Gamma_i),$
where $h(\Gamma_i)$ is the combinatorial Cheeger constant. The analogue of Cheeger-Buser for graphs leads to
$h(\Gamma_i)\gg \lambda_1(\Gamma_i),$
and if we know the esperanto property
$\lambda_1(\Gamma_i)\gg 1/(\log d_i)^A$
for some fixed $A\geq 0$ , we see that the logarithmic decay of the Cheeger constant is more than compensated by the growth of the volume in
$d(K_i)\gg h_i V_i.$

(Note: the proof of the main inequality (*) in the paper of Gromov and Guth is quite delicate and is much deeper and more sophisticated mathematics than the rather formal considerations above! I hope to be able to understand more of it in the coming weeks and months — indeed, the amount and variety of ideas and tools in this paper is quite remarkable…)