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In mathematics, specifically in the field of group theory, the McKay conjecture is a conjecture of equality between the number of irreducible complex characters of degree not divisible by a prime number ${\displaystyle p}$ to that of the normalizer of a Sylow ${\displaystyle p}$-subgroup. It is named after Canadian mathematician John McKay.

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 ${\displaystyle G}$.

History

In McKay's original papers on the subject, 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 ${\displaystyle G}$. 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.

Proof

In 2007, Marty 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. This opens the door to a proof of the conjecture by using the classification of finite simple groups.

The paper of Isaacs-Malle-Navarro was also an inspiration for similar reductions for Alperin weight conjecture, its block version, the Alperin-McKay conjecture and Dade's conjecture.

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

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 in the form of an ${\displaystyle {\textrm {Aut}}(G)}$-equivariant Jordan decomposition of characters for finite quasisimple groups of Lie type.

The McKay conjecture for all primes and all finite groups was announced by Marc Cabanes and Britta Späth in October 2023 in various conferences, a manuscript being available later in 2024.

References

1. McKay, John (1971). "A new invariant for finite simple groups". Notices of the American Mathematical Society. 128: 397.
2. McKay, John (1972). "Irreducible representations of odd degree". Journal of Algebra. 20: 416–418. doi:10.1016/0021-8693(72)90066-X.
3. Isaacs, I. Martin (1973). "Characters of solvable and symplectic groups". American Journal of Mathematics. 95: 594–635. doi:10.2307/2373731.
4. 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.
5. 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.
6. 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.
7. Malle, Gunter; Späth, Britta (2016). "Characters of odd degree". Annals of Mathematics. 184: 869–908. doi:10.4007/annals.2016.184.3.6.
8. Späth, Britta (2023). "Extensions of characters in type D and the inductive McKay condition, II". arXiv:2304.07373 .
9. Marc Cabanes; Britta Späth (2024). "The McKay Conjecture on character degrees". arXiv:2410.20392 .

• Representation theory of groups