Alperin weight bijections and dimensions of simple modules

Abstract

We propose a new refinement of the (blockwise) Alperin weight conjecture taking dimensions of simple modules (or degrees of irreducible Brauer characters) into consideration. This is similar to Giannelli's refinement of the (Alperin–)McKay conjecture. We prove this new refinement for blocks with a cyclic or Klein four defect group, for inertial blocks (which include the Broué–Puig nilpotent blocks), and for $p$-blocks of $p$-solvable groups.