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 ${\displaystyle p}$ and the order of the normalizer 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 Gunter Malle and Britta Späth in 2016. ^[8]

An important step in proving the inductive McKay condition for all simple groups is to determine the action of the group of automorphisms ${\displaystyle {\textrm {Aut}}(G)}$ 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]

References

1. Sloman, Leila (2025-02-19). "After 20 Years, Math Couple Solves Major Group Theory Problem". Quanta Magazine. Retrieved 2025-02-20.
2. McKay, John (1971). "A new invariant for finite simple groups". Notices of the American Mathematical Society . 128: 397.
3. McKay, John (1972). "Irreducible representations of odd degree". Journal of Algebra . 20: 416–418. doi:10.1016/0021-8693(72)90066-X.
4. Isaacs, I. Martin (1973). "Characters of solvable and symplectic groups". American Journal of Mathematics . 95: 594–635. doi:10.2307/2373731.
5. Alperin, Jon L. (1976). "The main problem in block theory". Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975). Academic Press. pp. 341–356. ISBN   978-3-540-20364-3.
6. Isaacs, I. M.; Malle, Gunter; Navarro, Gabriel (2007). "A reduction theorem for the McKay conjecture". Inventiones Mathematicae . 170: 33–101. doi:10.1007/s00222-007-0057-y.
7. Navarro, Gabriel (2018). Character theory and the McKay conjecture. Cambridge Studies in Advanced Mathematics. Vol. 175. Cambridge University Press. ISBN   978-1-108-42844-6.
8. Malle, Gunter; Späth, Britta (2016). "Characters of odd degree". Annals of Mathematics . 184: 869–908. doi:10.4007/annals.2016.184.3.6.
9. Späth, Britta (2023). "Extensions of characters in type D and the inductive McKay condition, II". arXiv: 2304.07373 [RT].
10. Marc Cabanes; Britta Späth (2024). "The McKay Conjecture on character degrees". arXiv: 2410.20392 [RT].