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.