Draft:Brauer height zero conjecture

In mathematics, specifically in the field of modular representation theory, the height zero conjecture is a conjecture published by Richard Brauer in 1956 . It aims at relating the degrees of characters in a $p$-block $B$ with the structure of the defect groups of $B$.

Statement

Fix $p$ a prime number. For $n$ a positive integer, let $n=n_p{n}_{p^{\prime }}$ where $n_p$ is a power of $p$ and $p\nmid n_{p^{\prime }}$.

Let $G$ be a finite group, and $B$ a $p$-block of $G$. Let $D\le G$ be a defect group of $B$. Brauer has shown that every ordinary irreducible character in $B$ is such that its degree satisfies

$\chi (1{)}_p=\frac{|G{|}_p}{|D|}{p}^h$

for some integer $h\ge 0$. The integer $h$ is called the height of $\chi$ in its $p$-block. Every block has ordinary irreducible characters of height zero.

The statement of Brauer's height zero conjecture (BHZ) is as follows.

All ordinary characters of $G$ in $B$ have height $h=0$ if and only if $D$ is abelian.

It is costumary to abbreviate the "if" part as (BHZ1) and the "only if" part as (BHZ2).

Proof

The conjecture was checked for solvable groups by Brauer's student Paul Fong in 1960.

In 1988 Thomas R. Berger and Reinhard Knörr showed that BHZ1 for a given prime is equivalent to the same statement for quasisimple groups only. The checking of all quasisimple groups was achieved by Radha Kessar and Gunter Malle in 2013, thus establishing BHZ1.

In 2024, Gunter Malle, Gabriel Navarro, A. A. Schaeffer Fry and Pham Huu Tiep proved the other half, the "only if" part of the Brauer height zero conjecture for odd primes $p$. This part of the Brauer height zero conjecture for the prime 2 was proven by Lucas Ruhstorfer by other methods along with the Alperin-McKay conjecture for that prime number. This finished the proof of the Brauer height zero conjecture for all finite groups and prime numbers.