Brauer's height zero conjecture

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

Let ${\displaystyle \nu }$ be the discrete valuation defined on the integers by ${\displaystyle \nu (mp^{a})=a}$ where ${\displaystyle m}$ is coprime to ${\displaystyle p}$. Brauer proved that if ${\displaystyle B}$ is a block with defect group ${\displaystyle D}$ then ${\displaystyle \nu (\chi (1))\geq \nu (|G:D|)}$ for each ${\displaystyle \chi \in \mathrm {Irr} (B)}$. Brauer's Height Zero Conjecture asserts that ${\displaystyle \nu (\chi (1))=\nu (|G:D|)}$ for all ${\displaystyle \chi \in \mathrm {Irr} (B)}$ if and only if ${\displaystyle D}$ 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 ${\displaystyle G}$ determines if its Sylow ${\displaystyle p}$-subgroups are abelian. Solving Brauer's height zero conjecture for blocks whose defect groups are Sylow ${\displaystyle p}$-subgroups (or equivalently, that contain a character of degree coprime to ${\displaystyle p}$) 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 ${\displaystyle p}$-solvable groups by David Gluck and Thomas R. Wolf.^[5] The so-called generalized Gluck—Wolf theorem, which was a main obstacle towards a proof of 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 ${\displaystyle p=2}$.^[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]