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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 $\mathrm {Irr} (B)$ .

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 $\chi \in \mathrm {Irr} (B)$ . Brauer's Height Zero Conjecture asserts that $\nu (\chi (1))=\nu (|G:D|)$ for all $\chi \in \mathrm {Irr} (B)$ 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 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 $p=2$ .^[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]

References

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