Brauer's height zero conjecture explained

The Brauer Height Zero Conjecture is a conjecture in modular representation theory of finite groups relating the degrees of the complex irreducible characters in a Brauer block and the structure of its defect groups. It was formulated by Richard Brauer in 1955.

Statement

Let

G be a finite group and p a prime. The set {\rm Irr}(G) of irreducible complex characters can be partitioned into Brauer p -blocks. To each p -block B is canonically associated a conjugacy class of p -subgroups, called the defect groups of B . The set of irreducible characters belonging to B is denoted by Irr (B) .

Let

\nu be the discrete valuation defined on the integers by \nu(mp ^{a)=a} where m is coprime to p . Brauer proved that if B is a block with defect group D then \nu(\chi( 1 ))\geq \nu(|G:D|) for each \chi\in Irr (B) . Brauer's Height Zero Conjecture asserts that \nu(\chi( 1 )) = \nu(|G:D|) for all \chi\in Irr (B) if and only if D is abelian.

History

Brauer's Height Zero Conjecture was formulated by Richard Brauer in 1955.^[1] It also appeared as Problem 23 in Brauer's list of problems.^[2] Brauer's Problem 12 of the same list asks whether the character table of a finite group

G determines if its Sylow p -subgroups are abelian. Solving Brauer's height zero conjecture for blocks whose defect groups are Sylow p -subgroups (or equivalently, that contain a character of degree coprime to p ) also gives a solution to Brauer's Problem 12.

Proof

The proof of the if direction of the conjecture was completed by Radha Kessar and Gunter Malle^[3] in 2013 after a reduction to finite simple groups by Thomas R. Berger and Reinhard Knörr.^[4]

The only if direction was proved for

p -solvable groups by David Gluck and Thomas R. Wolf.^[5] The so called generalized Gluck—Wolf theorem, which was a main obstacle towards a proofof the Height Zero Conjecture was proven by Gabriel Navarro and Pham Huu Tiep in 2013.^[6] Gabriel Navarro and Britta Späth showed that the so-called inductive Alperin—McKay condition for simple groups implied Brauer's Height Zero Conjecture.^[7] Lucas Ruhstorfer completed the proof of these conditions for the case p =2 .^[8] The case of odd primes was finally settled by Gunter Malle, Gabriel Navarro, A. A. Schaeffer Fry and Pham Huu Tiep using a different reduction theorem.^[9]

Notes and References

1. Book: Brauer . Richard D. . 1956 . Proceedings of the International Symposium on Algebraic Number Theory, Tokyo and Nikko, 1955 . Number theoretical investigations on groups of finite order . Science Council of Japan. 55–62.
2. Book: Brauer . Richard D. . 1963 . Representations of finite groups. Lectures in Mathematics. 1 . Wiley . 133-175 .
3. Kessar. Radha. Malle. Gunter. Quasi-isolated blocks and Brauer's height zero conjecture. 10.4007/annals.2013.178.1.6 . . 178 . 2013. 321-384. 1112.2642.
4. Berger . Thomas R. . Knörr . Reinhard . On Brauer's height 0 conjecture . . 109 . 1988 . 109–116 . 10.1017/S0027763000002798 .
5. Gluck. David. Wolf. Thomas R.. Brauer's height conjecture for p-solvable groups. Transactions of the American Mathematical Society. 282. 1984. 137–152. 10.2307/1999582.
6. Navarro . Gabriel . Tiep . Pham Huu . Characters of relative p' -degree over normal subgroups . . 178 . 2013 . 1135–1171 . 10.4007/annals.2013.178.
7. Navarro. Gabriel. Späth. Britta. On Brauer's height zero conjecture. Journal of the European Mathematical Society. 16. 2014. 695–747. 10.4171/JEMS/444. 2209.04736.
8. Ruhstorfer. Lucas. 2022. The Alperin-McKay conjecture for the prime 2. to appear in Annals of Mathematics.
9. Malle, Gunter. Navarro, Gabriel. Schaeffer Fry, A. A.. Tiep, Pham Huu. Brauer's Height Zero Conjecture. 2024. 10.4007/annals.2024.200.2.4 . 200 . . 557-608. 2209.04736.