Weights for 𝜋-partial characters of 𝜋-separable groups

Abstract The aim of this paper is to confirm a conjecture of Isaacs and Navarro from 1995, which asserts that, for any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>π</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> \pi^{\prime} -subgroup 𝑄 of a 𝜋-separable group 𝐺, the number of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>π</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> \pi^{\prime} -weights of 𝐺 with 𝑄 as the first component is larger than or equal to the number of irreducible 𝜋-partial characters of 𝐺 with 𝑄 as their vertex. We also give a sufficient condition to guarantee that these two numbers are equal, and thereby strengthen their main theorem on the 𝜋-version of the Alperin weight conjecture.