McKay conjecture

In mathematics, specifically in the field of group theory, the McKay conjecture is a theorem of equality between two numbers: the number of irreducible complex characters of degree not divisible by a prime number ${\displaystyle p}$ for a given finite group and the same number for the normalizer in that group of a Sylow ${\displaystyle p}$-subgroup.

It is named after the Canadian mathematician John McKay, who originally stated a limited version of it as a conjecture in 1971, for the special case of ${\displaystyle p=2}$ and simple groups. The conjecture was later generalized by other mathematicians to a more general conjecture for any prime value of ${\displaystyle p}$ and more general groups.

In 2023, a proof of the general conjecture was announced by Britta Späth and Marc Cabanes.^[1]

Statement

Suppose ${\displaystyle p}$ is a prime number, ${\displaystyle G}$ is a finite group, and ${\displaystyle P\leq G}$ is a Sylow ${\displaystyle p}$-subgroup. Define

${\displaystyle {\textrm {Irr}}_{p'}(G):=\{\chi \in {\textrm {Irr}}(G):p\nmid \chi (1)\}}$

where ${\displaystyle {\textrm {Irr}}(G)}$ denotes the set of complex irreducible characters of the group ${\displaystyle G}$. The McKay conjecture claims the equality

${\displaystyle |{\textrm {Irr}}_{p'}(G)|=|{\textrm {Irr}}_{p'}(N_{G}(P))|}$

where ${\displaystyle N_{G}(P)}$ is the normalizer of ${\displaystyle P}$ in ${\displaystyle G}$.

In other words, for any finite group ${\displaystyle G}$, the number of its irreducible complex representations whose dimension is not divisible by ${\displaystyle p}$ equals that number for the normalizer of any of its Sylow ${\displaystyle p}$-subgroups. (Here we count isomorphic representations as the same.)

History

In McKay's original papers on the subject,^[2][3] the statement was given for the prime ${\displaystyle p=2}$ and simple groups, but examples of computations of ${\displaystyle |{\textrm {Irr}}_{p'}(G)|}$ for odd primes or symmetric groups are mentioned. Marty Isaacs also checked the conjecture for the prime 2 and solvable groups ${\displaystyle G}$.^[4] The first appearance of the conjecture for arbitrary primes is in a paper by Jon L. Alperin giving also a version in block theory, now called the Alperin–McKay conjecture.^[5]

Proof

In 2007, Martin Isaacs, Gunter Malle and Gabriel Navarro showed that the McKay conjecture reduces to the checking of a so-called inductive McKay condition for each finite simple group.^[6][7] This opens the door to a proof of the conjecture by using the classification of finite simple groups.

The Isaacs−Malle−Navarro paper was also an inspiration for similar reductions for Alperin weight conjecture (named after Jonathan Lazare Alperin), its block version, the Alperin−McKay conjecture, and Dade's conjecture (named after Everett C. Dade).

The McKay conjecture for the prime 2 was proven by Britta Späth and Gunter Malle in 2016.^[8]

An important step in proving the inductive McKay condition for all simple groups is to determine the action of the automorphism group ${\displaystyle {\textrm {Aut}}(G)}$ on the set ${\displaystyle {\textrm {Irr}}(G)}$ for each finite quasisimple group ${\displaystyle G}$. The solution has been announced by Späth^[9] in the form of an ${\displaystyle {\textrm {Aut}}(G)}$-equivariant Jordan decomposition of characters for finite quasisimple groups of Lie type.

A proof of the McKay conjecture for all primes and all finite groups was announced by Britta Späth and Marc Cabanes in October 2023 in various conferences, a manuscript being available later in 2024.^[10]