On the projective height zero conjecture

Abstract

Recently, Malle and Navarro put forward a projective version of Brauer's celebrated height zero conjecture on blocks of finite groups. In this short note we show that Brauer's original conjecture implies the projective version.The following is a long-standing conjecture in representation theory of finite groups: Conjecture 1 (Brauer's height zero conjecture [2, Problem 23]). Let B be a block of a finite group with defect D. Then every irreducible character in B has height 0 if and only if D is abelian. Recently, Malle-Navarro [6] proposed the following generalization of Conjecture 1 (the case Z = 1 yields the original conjecture). An equivalent statement in terms of θ-blocks was given by Rizo [9]. Conjecture 2 (Malle-Navarro's projective height zero conjecture). Let B be a p-block of a finite group G with defect group D. Let Z be a central p-subgroup of G and let λ ∈ Irr(Z). Then every irreducible character in B lying over λ has height 0 if and only if D/Z is abelian and λ extends to D.