Character Theory and the McKay Conjecture

What does the McKay Conjecture say? In the preface of the book under review, Gabriel Navarro leads us to the answer as follows:

In 1972, in a paper dedicated to Richard Brauer, J. McKay observed that in some simple groups the number of irreducible complex characters of odd degree was equal to the same number calculated for the group \(N_G(P)\), the normalizer of a Sylow \(2\)-subgroup \(P\) of \(G\). Of course the groups \(G\) and \(N_G(P)\) are very different in general, and yet they seemed to share a fundamental invariant. If true, this was an unexplained and astonishing discovery. A year later, I. M. Isaacs proved that McKay’s observation was true for solvable groups … and for groups of odd order for every prime. Although never formally formulated in its full generality, the McKay Conjecture took form … [To wit:] If \(G\) is a finite group, \(p\) is a prime and \(m_p(G)\) is the number of irreducible complex characters of \(G\) of degree not divisible by \(p\), then \(m_p(G)=m_p(N_G(P))\) where \(P\in \mathrm{Syl}_p(G)\).

Navarro continues his informative preface with an exhaustive overview of the work done on this conjecture, and he includes a number of fascinating insights and elaborations. We learn, for example, of a 1975 generalization of the conjecture due to J. L. Alperin and proofs of the assertion’s generalization for symmetric groups and -solvable groups. In connection with the latter, Navarro makes the following pithy if amusing observation:

… different families of groups were checked by many mathematicians, and soon the idea that the McKay conjecture was going to prove correct was generally accepted. As Alperin used to say informally, if instead of mathematics this were physics, the McKay model would have been accepted long ago.

But happily it isn’t physics , so we read the following:

During the course of the years, several deep generalizations of the conjecture have provided hints about the ingredients of a possible proof: [Brauer] blocks, isometries, simplicial complexes, characters over the \(p\)-adics, derived categories: some, or all, of these should or could be involved in the proof … but nobody can figure out exactly how.

We are dealing with a subject that is very much alive, in the sense of it still being unsettled, and a lot of work is being done in an attempt to get the prize. Most notably,

[i]n 2007, G. Malle, I. M. Isaacs and [Navarro] published a reduction of the McKay conjecture to a problem on simple groups. This reduction was specially tailored for the McKay problem … This is the … so-called inductive McKay condition … It was proved that the McKay conjecture [is] true for every finite group if every simple group satisfies this inductive McKay condition.

So we have here a very strong result indeed, and a promising approach; it hearkens back to the so-called CFSG, which, yes, is the vaunted Classification of Finite Simple Groups. The author goes on to say,

Since 2007, there have been significant simplifications on the formulation of the inductive McKay condition on quasisimple groups, and a general theory on this, with implications for the representation theory of general finite groups, has been developed (mainly by B. Späth). We shall dedicate the last chapter of this book to this topic. We wish not only to publicize the beauty of this reduction but also to engage students in the exciting tasks that remain ahead of us.

And now we know what the goal of this book is. It is very serious (finite) group theory, with representation (and character) theory playing a huge role, and it is aimed at making converts to the cause. The book looks to be a compact and complete presentation of all the ingredients that go into a solid education for such aspiring scholars, and of course an expert account of the material Navarro described above surrounding the McKay Conjecture (and then some). The early chapters of the book deal with character theory on anabolic steroids — all very interesting material, e.g. “Galois actions on characters” (very tantalizing) and a lot of Brauer theory. And one encounters a number of classics along the way: Burnside’s theorem and stuff on normal Sylow \(p\)-subgroups (the comfort of a familiar theme from (under)graduate school, but now it’s all souped up of course). And then in the last two chapters we get to material alluded to above, at the frontier of research on the McKay Conjecture.

It's group theory, group theory, and more group theory, modulo the understanding that character theory is part of … group theory.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.