Character Theory and the McKay Conjecture
by Gabriel Navarro
Cambridge University Press, 2018
Reviewed by Gerald Cliff, University of Alberta
To state McKay’s Conjecture, for a prime p and a finite group G, let mp(G) denote the number of irreducible complex characters of G whose degree is not divisible by p. Let NG(P) denote the normalizer of a Sylow p-subgroup of G. The conjecture is that
mp(G) = mp(NG(P )).
This conjecture was made in the early 1970s, and has become one of the main problems in the representation theory of finite groups. In the 2000s, an effort was made by Navarro and collaborators to reduce this problem to the case that G is a finite simple group, and then use the classification of finite simple groups. There is a stronger conjecture which implies McKay’s, and which would hold if it holds for all finite simple groups. At this time it is not known that the stronger conjecture does indeed hold for all finite simple groups, except for p = 2, so that McKay’s conjecture is true for p = 2.
In this book, the author gives a good presentation of the theory of characters of finite groups, including some recent interesting results. He shows how to reduce the stronger conjecture to simple groups. The book could be read by graduate students and non-experts.
The Best Writing on Mathematics, 2019
Edited by Mircea Pitici
Princeton University Press, 2019
Reviewed by Karl Dilcher
This is the tenth volume in a remarkable series of annual anthologies. A year ago in this space I addressed some general features shared by all volumes. I will not repeat these remarks here; the interested reader will find them in the September 2019 issue. Instead, I will quote from the overview of this volume: