John McKay

I learned only yesterday that John McKay died a month ago.

John McKay made the kind of contribution to the subject that most mathematicians can only dream of, in part because of his wide-ranging interests and keen observation. He was also a pioneer of the use of computers in mathematics: he and Graham Higman constructed two of the sporadic finite simple groups, Janko’s third group and Held’s group, in the early 1970s.

John is best known for two stunning observations.

The one with the most impact, and the easiest to state, was probably noticing that the degree of the smallest non-trivial linear representation of the Monster (which is 196883) is one less than the first non-trivial Fourier coefficient of the classical modular function. (Googling “196884=196883+1” gives 1200 hits.) It was remarked at the time that few mathematicians were familiar with both of these topics, which appear at first sight to be completely unconnected. John Conway named this monstrous moonshine, and it led to connections with vertex operator algebras and a Fields medal for Richard Borcherds.

The second big conjecture concerned the binary rotation groups (that is, the double covers of the finite rotation groups in Euclidean 3-space obtained by pulling them back to the 2-dimensional complex unitary group by the two-to-one homomorphism to the real 3-dimensional orthogonal group. Such a group has a natural 2-dimensional unitary representation χ. If we form a graph whose vertices are the irreducible representations, each vertex labelled with the degree, then the result is connected, and the sum of the labels on the neighbours of a vertex is twice the label of a vertex. Thus the graph has greatest eigenvalue 2, and is an extended Dynkin diagram of type ADE. I have described this in my series about the ADE affair.

But there is more. I am currently at a workshop at the Isaac Newton Institue on Counting conjectures and beyond in representation theory. Another observation of John McKay was at the origin of this field: he observed that, in the simple groups G he examined, if p is prime, then the number of irreducible representations of G of degree coprime to p is equal to the corresponding number for the normaliser of a Sylow p subgroup of G. The conjecture was later extended to all finite groups, and a lot of progress has been made, in particular, the conjecture is true for soluble groups. It has also taken its place among several further conjectures with a similar “local-to-global” form. However, I know much less about this, so I will say no more.