A reduction theorem for the blockwise Navarro Alperin weight conjecture via H-triples

TL;DR

This paper refines the reduction of the Galois Alperin weight conjecture by introducing a blockwise version and establishing a stronger form via H-triples under certain conditions.

Contribution

It proposes the blockwise Galois Alperin weight conjecture and links its validity to the original conjecture through H-triples and isomorphisms.

Findings

Refined reduction of GAW conjecture to blockwise version.

Established stronger conjecture versions assuming inductive conditions.

Connected conjectures via central and block isomorphisms of H-triples.

Abstract

The Galois Alperin weight (GAW) conjecture has been reduced to the inductive GAW condition for simple groups. We proceed in two steps to refine this reduction. First, we propose the blockwise Galois Alperin weight (BGAW) conjecture and define its associated inductive BGAW condition. Second, assuming the inductive GAW (respectively, BGAW) condition for simple groups, we establish a stronger version of the GAW (respectively, BGAW) conjecture in terms of central (respectively, block) isomorphism of H-triples.