McKay conjecture
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 to that of the normalizer of a Sylow -subgroup. It is named after Canadian mathematician John McKay.
Statement
[edit]Suppose is a prime number, is a finite group, and is a Sylow -subgroup. Define where denotes the set of complex irreducible characters of the group . The McKay conjecture claims the equality where is the normalizer of in .
History
[edit]In McKay's original papers on the subject [1][2], the statement was given for the prime but examples of computations of for odd primes are also mentioned. 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 [3]
Proof
[edit]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 [4][5]. 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 [6].
An important step in proving the inductive McKay condition for all simple groups is to determine the action of the group of automorphisms on the set for each finite quasisimple group . The solution has been announced by Späth in [7].
The McKay conjecture for all primes and all finite groups was announced by Marc Cabanes and Britta Späth in October 2023.
References
[edit]- ^ McKay, John (1971). "A new invariant for finite simple groups". Notices of the American Mathematical Society. 128: 397.
- ^ McKay, John (1972). "Irreducible representations of odd degree". Journal of Algebra. 20: 416–418. doi:10.1016/0021-8693(72)90066-X.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ Malle, Gunter; Späth, Britta (2016). "Characters of odd degree". Annals of Mathematics. 184: 869–908. doi:10.4007/annals.2016.184.3.6.
- ^ Späth, Britta (2023). "Extensions of characters in type D and the inductive McKay condition, II". arXiv:2304.07373 [RT].