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.
of irreducible complex characters can be partitioned into Brauer
is canonically associated a conjugacy class of
-subgroups, called the defect groups of
The set of irreducible characters belonging to
be the discrete valuation defined on the integers by
Brauer proved that if
is a block with defect group
Brauer's Height Zero Conjecture asserts that
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
Solving Brauer's height zero conjecture for blocks whose defect groups are Sylow
-subgroups (or equivalently, that contain a character of degree coprime to
) also gives a solution to Brauer's Problem 12.
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.
-solvable groups by David Gluck and Thomas R.
[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
[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.