The McKay conjecture was refined in 2004 by G. Navarro 
taking into account the action of Galois automorphisms on characters. 
This refinement is known as the Galois McKay conjecture, and it is one 
of the last counting conjectures without a reduction theorem. In this 
talk I will present a possible reduction theorem for the Galois McKay 
conjecture to a question on finite simple groups. This is joint work 
with G. Navarro and B. Späth.

 

The Navarro conjecture states that the actions of a particular subgroup of Galois automorphisms on the two sets of characters involved in the McKay conjecture should be permutation isomorphic. This conjecture predicted that the local condition that a Sylow $p$-subgroup $P$ of a finite group $G$ is self-normalizing can be characterized in terms of the character theory of $G$; a prediction that has been recently verified for all primes. In the same spirit, we restrict our attention to the character theory of the principal $p$-block and analyze the relation with the structure of ${\bf N}_G(P)$.