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 $p$ for a given finite group and the same number for the normalizer in that group of a Sylow $p$ -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 $p=2$ and simple groups. The conjecture was later generalized by other mathematicians to a more general conjecture for any prime value of $p$ and more general groups.

In 2023, a proof of the general conjecture was announced by Britta Späth and Marc Cabanes.{{Cite web |last=Sloman |first=Leila |date=2025-02-19 |title=After 20 Years, Math Couple Solves Major Group Theory Problem |url=https://www.quantamagazine.org/after-20-years-math-couple-solves-major-group-theory-problem-20250219/ |access-date=2025-02-20 |website=Quanta Magazine |language=en}}

Statement

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

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

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

: $|\textrm{Irr}_{p'}(G)| = |\textrm{Irr}_{p'}(N_G(P))|$

where $N_G(P)$ is the normalizer of $P$ in $G$ .

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

History

In McKay's original papers on the subject,{{cite journal |last=McKay |first=John |date=1971 |title=A new invariant for finite simple groups|volume=128 |journal=Notices of the American Mathematical Society |pages=397}}

{{cite journal |last=McKay |first=John |date=1972 |title=Irreducible representations of odd degree |doi=10.1016/0021-8693(72)90066-X |volume=20 |journal=Journal of Algebra |pages=416–418}} the statement was given for the prime $p=2$ and simple groups, but examples of computations of $|\textrm{Irr}_{p'}(G)|$ for odd primes or symmetric groups are mentioned. Marty Isaacs also checked the conjecture for the prime 2 and solvable groups $G$ .

{{cite journal |last=Isaacs |first=I. Martin |date=1973 |title=Characters of solvable and symplectic groups |doi=10.2307/2373731 |volume=95 |journal=American Journal of Mathematics |pages=594–635}} 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.{{cite book

| last1 = Alperin | first1 = Jon L.

| year = 1976

| title = Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975)

| chapter = The main problem in block theory

| publisher=Academic Press

| pages=341–356

| isbn = 978-3-540-20364-3

}}

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.{{cite journal |last1=Isaacs |first1= I. M. |last2=Malle |first2=Gunter|last3=Navarro|first3=Gabriel |date=2007 |title=A reduction theorem for the McKay conjecture|doi=10.1007/s00222-007-0057-y |volume=170 |journal=Inventiones Mathematicae |pages=33-101}}

{{cite book

| last = Navarro | first = Gabriel

| year = 2018

| title = Character theory and the McKay conjecture

| series = Cambridge Studies in Advanced Mathematics

| volume = 175

| publisher = Cambridge University Press

| isbn =978-1-108-42844-6

}} 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.{{cite journal |last1=Malle |first1= Gunter |last2=Späth |first2=Britta|date=2016 |title=Characters of odd degree|doi=10.4007/annals.2016.184.3.6 |volume=184 |journal=Annals of Mathematics |pages=869-908}}

An important step in proving the inductive McKay condition for all simple groups is to determine the action of the automorphism group $\textrm{Aut}(G)$ on the set $\textrm{Irr}(G)$ for each finite quasisimple group $G$ . The solution has been announced by Späth{{cite arXiv

| last = Späth

| first = Britta

| date = 2023

| title = Extensions of characters in type D and the inductive McKay condition, II

| eprint = 2304.07373

| class = RT

}} in the form of an $\textrm{Aut}(G)$ -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.{{cite arxiv | eprint=2410.20392 | title=The McKay Conjecture on character degrees | author1=Marc Cabanes | author2=Britta Späth | class= RT |year=2024}}to appear Annals of Mathematics

References

Category:Representation theory of groups

Category:Conjectures that have been proved