Well, that break was longer than intended…

In the last post, we claimed that $\displaystyle {\rm AM}_{\rm Z}(G)>1$ for every finite, non-abelian group G. It turns out that the easiest way to prove this goes via a certain minorant for ${\rm AM}_{\rm Z}(G)$ which we will work with in some subsequent posts. In this post, we’ll introduce this minorant, give an explicit lower bound, and then briefly indicate how it allows us to show the stronger result that

$\displaystyle \inf \{ {\rm AM}_{\rm Z}(G) \colon G \mbox{ finite and non-abelian} \} > 1\;.$

1. Recap

Recall that

$\displaystyle {\rm AM}_{\rm Z}(G) = \sum_{C,D\in{\rm Conj}(G)} |C|\ |D| \left\vert \sum_{\phi\in {\rm Irr}(G)} \frac{1}{|G|^2} \phi(e)^2\phi(C)\phi(D) \right\vert \;.$

We can rewrite this in a cosmetic but suggestive way. Observe that the inversion map on G, which sends each element to its inverse, maps conjugacy classes to conjugacy classes. It follows that for each D in Conj(G), the set

$\displaystyle \overline{D} = \{ x^{-1} \colon x\in D \}$

also belongs to Conj(G). Moreover, the map ${D \mapsto \overline{D}}$ is an involution, in particular is bijective. Therefore, since ${\phi(\overline{D})=\overline{\phi(D)}}$ for every character ${\phi}$, we obtain

$\displaystyle {\rm AM}_{\rm Z}(G) = \sum_{C,D\in{\rm Conj}(G)} |C|\ |D| \left\vert \sum_{\phi\in {\rm Irr}(G)} \frac{1}{|G|^2} \phi(e)^2\phi(C)\overline{\phi(D)} \right\vert \;.$

We already saw this idea, in a special case, when we looked at ${{\rm AM}_{\rm Z}(G)}$ for abelian groups. There, the point of this small change was that it made the expression look more like an inner product, so that one could apply Schur orthogonality relations; a similar idea was applied in a recent paper of Alaghmandan, Samei and myself (arXiv 1302.1929) to handle certain groups which are close to the abelian case in some sense.

2. A remark on normalized versus unnormalized counting measure

First, I need to clear up an issue of normalization conventions, which I omitted to deal with before. In our series of posts, we have always been working on the complex group algebra equipped with the ${\ell^1}$-norm. That is, we are looking at ${L^1(G,\lambda)}$ where ${\lambda}$ denotes counting measure on the finite set G.

On the other hand, the paper of Azimifard–Samei–Spronk (henceforth referred to as [ASS09]), where the amenability constant of the centre of the group algebra was first studied, considers ${L^1(G,\mu)}$ where G is a compact group and ${\mu}$ denotes uniform probability measure on G.

However, there is no serious conflict. For if G is a finite group, let A denote ${\ell^1(G)}$ equipped with counting measure ${\lambda}$ and equipped with convolution using ${\lambda}$, and let B denote ${\ell^1(G)}$ equipped with uniform probability measure ${\mu}$ and equipped with convolution using ${\mu}$. Then a direct calculation shows that the obvious isometric rescaling map from A to B is in fact an isomorphism of Banach algebras. In particular, A and B have the same amenability constant. Thus, our formula from ${{\rm AM}_{\rm Z}(G)}$ coincides with the formula in [ASS09] for the amenability constant of ${L^1(G,\mu)}$.

3. A minorant for ${{\rm AM}_{\rm Z}(G)}$

At a naive level (but not a completely facile one) we might say that the difficulty in getting non-trivial lower bounds on ${{\rm AM}_{\rm Z}(G)}$ is due to the fact that one takes the modulus of a sum of different terms, inside which there might be significant cancellation. Indeed, this is exactly what happens in the case of an abelian group: see the previous post for details.

One situation where we can avoid cancellation is where the terms in the sum are all non-negative, so that the modulus is just the sum itself. Looking at the revised formula for ${{\rm AM}_{\rm Z}(G)}$, we see that this happens whenever C=D (it may also happen for some other choices of C and D, but let us ignore that for now). Moreover, if we only want a lower bound on ${{\rm AM}_{\rm Z}(G)}$ and not its precise value, we are free to discard terms indexed by particular C and D. Thus, as observed in [ASS09], ${{\rm AM}_{\rm Z}(G)}$ is bounded below by the following quantity

\displaystyle \begin{aligned} \alpha(G) & := \sum_{C\in{\rm Conj}(G)} |C|^2 \left\vert \sum_{\phi\in {\rm Irr}(G)} \frac{1}{|G|^2} \phi(e)^2\phi(C)\overline{\phi(C)} \right\vert \\ & = |G|^{-2} \sum_{C\in{\rm Conj}(G)} \sum_{\phi\in {\rm Irr}(G)} \phi(e)^2 |\phi(C)|^2 |C|^2 \end{aligned} \ \ \ \ \ (1)

(The paper [ASS09] does not give this quantity a specific symbol, but in subsequent posts it will appear frequently enough that some extra notation seems warranted.)

In the previous post, we claimed that if G is a non-abelian finite group then we have ${\rm AM}_{\rm Z}(G)$ > 1. We can now give a sharper statement. (The calculation in [ASS09] does not give the explicit bound that we do, but it is implicit in their work.)

Proposition 1 (Azimifard–Samei–Spronk, 2009) Let G be a finite, non-abelian group, and let

\displaystyle \begin{aligned} s & =\min \{ |C| \colon C\in {\rm Conj}(G), |C|\neq 1 \} \\ & \equiv \min \{ |\mbox{conj. class of } x | \colon x \in G\setminus Z(G) \}.\end{aligned}

Then

$\displaystyle \alpha(G) \geq 1 + (s^2-s)|G|^{-2} > 1 \;.$

Proof: Compare the formula (1) which defines $\alpha(G)$ with

$\displaystyle |G|^{-2} \sum_{C\in{\rm Conj}(G)} \sum_{\phi\in {\rm Irr}(G)} \phi(e)^2 |\phi(C)|^2 |C| \ \ \ \ \ (2)$

Rearranging the sum and using the Schur row and column orthogonality relations, we see that (2) is equal to

$\displaystyle |G|^{-2} \sum_{\phi\in {\rm Irr}(G)} \phi(e)^2\sum_{C\in{\rm Conj}(G)} |\phi(C)|^2 |C| = |G|^{-1} \sum_{\phi\in {\rm Irr}(G)} \phi(e)^2 = 1.$

Hence

$\displaystyle \alpha(G)-1 = \sum_{C\in{\rm Conj}(G)} \sum_{\phi\in {\rm Irr}(G)} \phi(e)^2 |\phi(C)|^2 (|C|^2-|C|) \;.$

Now all of the terms on the right hand side are non-negative. Some of them may be zero (for instance, whenever C consists of just a single point, or whenver ${\phi(C)=0}$) but we can identify at least one strictly positive term. Namely, let ${C_0}$ be a conjugacy class of size s, and consider the trivial character ${\varepsilon}$ which takes the value 1 everywhere. Then

$\displaystyle \varepsilon(e)^2 |\varepsilon(C_0)|^2 (|C_0|^2-|C_0|) = s^2-s \geq 2,$

which gives us the lower bound that was claimed. $\hfill\Box$

Note that our lower bound “gets worse” as G gets bigger. Indeed, I believe the following question is still open.

Question. Is the infimum of $\alpha(G)$ over all finite non-abelian groups G strictly greater than 1?

Nevertheless, as mentioned in the first post of this series, we can do better when it comes to ${{\rm AM}_{\rm Z}(G)}$, which is the original quantity of interest. This was done in [ASS09] by appealing to a hard result of D. A. Rider, which tells us that the norms of central idempotents have “a gap at 1”.

Theorem 2 (Rider, 1973) Let K be a compact group, let E be a finite subset of Irr(K), and let ${f=\sum_{\phi\in E} \phi(e)\phi \in L^1(K)}$. (The orthogonality relations for irreducible characters imply that ${f}$ is a central idempotent in ${L^1(K)}$, and all central idempotents in ${L^1(K)}$ arise this way.) If ${\Vert f\Vert_1 > 1}$, then ${\Vert f \Vert_1 \geq 301/300}$.

Now let G be a finite, non-abelian group. Since ${\rm AM}_{\rm Z}(G)\geq \alpha(G)$, Proposition~1 immediately implies that ${{\rm AM}_{\rm Z}(G) > 1}$. Now ${{\rm AM}_{\rm Z}(G)=\Vert \Delta_G \Vert}$, where ${\Delta_G}$ is a central idempotent in ${L^1(G\times G)}$. Applying Rider’s theorem to ${\Delta_G}$ we deduce, as in [ASS09], that ${{\rm AM}_{\rm Z}(G)\geq 301/300}$.

Rider’s proof is rather long and technical and we will not present the details here. The constant 301/300 is somewhat arbitrary, resulting from choices made in chains of estimates, and can be improved slightly by repeating Rider’s arguments with more nit-picking. However, it seems that a significant improvement in the constant would require new ideas.

In the next a future post, we will see that with a more careful use of the Schur orthogonality relations, one can improve the lower bound in Proposition~1 to a constant that does not depend on |G|, provided that G has trivial centre. To do this we will need a new ingredient, not available in [ASS09], which ensures that a group which has an irreducible character of “surprisingly large” degree cannot have any small conjugacy classes except for elements of the centre.

Edited 2013-06-17: corrected some typos/omissions.

Edited 2014-12-30: revised rash promise.