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The central amenability constant of a finite group: Part 1 of n

6 May, 2013

Update 06/06/2013: see the bottom of the post (some typos also fixed)


As I am currently too besieged by a jumped-up virus to get any proper work done, I suppose I might as well put this blog to one of its nominal uses and actually talk about some mathematics I did recently (last Monday and Tuesday, in fact). Well, I say talk, I really mean “present” (and I am not at all sure I will manage to “explain”). Consider this my first experiment in using the blog for channeling rainwater

This post will just try to set up enough to state the result, and then if there is ever a follow-up post I will give more of the background story and the necessary details.


Fix a finite group G and look at the centre of its complex group algebra {\mathbb C}G. This algebra, which we call A for now, is commutative and spanned by its minimal idempotents. What are these minimal idempotents? they are scalar multiples of the irreducible characters of G. (A character of G is, for us, the trace of some representation over {\mathbb C}; we say the character is irreducible if the corresponding representation is.) If \phi is such a character then it is not hard to show that the corresponding idempotent in A is |G|^{-1}\phi(e)\phi.

Now an algebra such as A admits a unique so-called separating idempotent, that is, an element {\bf m} \in A\otimes A satisfying a\cdot {\bf m} = {\bf m}\cdot a for all a \in A and \Delta(m)=1_A, where \Delta:A\otimes A\to A is the linearization of the multiplication map. (This separating idempotent is a witness to the fact that A has trivial Hochschild cohomology, or trivial Ext if you prefer to think of it like that.) In fact it is not at all hard to work out what {\bf m} has to be:

\displaystyle{\bf m} = \sum_{\phi\in{\rm Irr(G)}} |G|^{-1}\phi(e)\phi \otimes |G|^{-1}\phi(e)\phi .

Let us accept for now that this element should be worth studying, and introduce norms. We can always equip the complex group algebra {\mathbb C}(G) with the \ell^1-norm, that is \Vert f\Vert = \sum_{x\in G} |f(x)|. The resulting Banach algebra \ell^1(G) is usually studied only for infinite G, but if one is interested in quantitative phenomena then the finite case still holds some mysteries and — I claim — some interest. In any case, since we can identify A\otimes A with the centre of the complex group algebra of G\times G, it inherits a natural norm as a subalgebra of \ell^1(G\times G).

We now define the central amenability constant of G to be the \ell^1-norm of the separating idempotent {\bf m}, and denote this number by {\rm AM}_{\rm Z}(G). If we wish to write things out explicitly, some more notation will be useful. Let {\rm Conj}(G) denote the set of conjugacy classes in G, and if C is a conjugacy class in G we write \phi(C) for the value \phi takes on one (hence on every) element of C. Then

\begin{aligned}   {\rm AM}_{\rm Z}(G)   & = \sum_{x,y\in G} \left\vert { \sum_{\phi \in {\rm Irr}(G)} |G|^{-2}\phi(e)^2 \phi(x) \phi(y) } \right\vert \\   & = \sum_{C,D\in {\rm Conj}(G)} |C| |D| \left\vert { \sum_{\phi \in {\rm Irr}(G)} |G|^{-2}\phi(e)^2 \phi(C) \phi(D) } \right\vert \,. \end{aligned}

This formula can be found1 as Theorem 1.8 of a 2009 JFA paper of Ahmadreza Azimifard, Ebrahim Samei and Nico Spronk (and before you complain about paywalls, this one is on the arXiv). At this point I should acknowledge that I first got interested in this invariant of G while listening to some talks given by Ebrahim about 5-6 years ago, and have since had a few enjoyable and educative discussions with both him and Nico on this topic.

A little patience (or some prior knowledge of Banach algebras) shows that if G is abelian then {\rm AM}_{\rm Z}(G)=1. Moreover, by an argument that will be outlined in the next installment, {\rm AM}_{\rm Z}(G)>1 whenever G is non-abelian. What is not all obvious, at least to me, is that 1 is an isolated value of this invariant. The following is extracted from the proof of Theorem 1.10 in the aforementioned paper.

Theorem (Azimifard–Samei–Spronk)
There exists δ > 0 such that {\rm AM}_{\rm Z}(G)\geq 1+\delta whenever G is non-abelian.

The authors obtain the existence of such a δ as a special case of a theorem of D. A. Rider (Transactions AMS, 1973). As an aside: for those of you who frequented MathStackExchange, this application is what ultimately prompted me to raise this question. What is rather unsatisfactory is how small the δ given by Rider’s proof is — Rider’s bounds yield δ = 1/300, although in some back of the envelope calculations I thought I could bring it down to around 1/90 — while the smallest value known for {\rm AM}_{\rm Z}(G) when G is nonabelian is 7/4.

After some failed attempts to get a better lower bound, I put the problem on the backburner, although from time to time it would come back to haunt me. More recently, in some joint work with Ebrahim and our PhD student Mahmood Alaghmandan, we had a closer look at the problem of calculating {\rm AM}_{\rm Z}(G) explicitly when you don’t know how to use GAP, and don’t have anyone at hand who does. See this preprint if you want to know more. In the process, I found myself getting interested again in the original problem of getting a lower bound closer to 7/4 than to 301/300…

Theorem (yours truly, some time last Monday)
{\rm AM}_{\rm Z}(G)\geq 4/3 whenever G is non-abelian.

This bound is still probably not sharp, but it does at least approach what seems to be the right value. However, more important than the improvement in the constant is that the argument now bypasses Rider’s theorem, which is much more general and whose proof I find somewhat hard to understand2. Instead of using general non-abelian Fourier analysis like Rider, who was in any case working on compact groups, we can get by with some basic arguments from character theory, together with a little structure theory for finite groups.

Well, this post looks long enough, so further explanation will have to wait for the next post…


Footnotes:

  1. I am cheating a little here, in that those authors have a slightly more conceptual definition of {\rm AM}_{\rm Z}(G), and have to do a little work to show it gives the same number as the formula above.
  2. One can follow the proof through line by line, but it leaves me rather baffled about what is conceptually going on, and what the intuition is.

Update 6th June 2013: it turns out that with a very modest tweak to some inequalities I was using, one can get a slight improvement from 4/3 to 13/9. More interestingly, some calculations in progress suggest that by using some group-theory and elbow-grease to deal with certain “edge cases”, one might be able to prove a lower bound of 7/4 (which we know would be sharp). Details to follow, perhaps …

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