Towards Donovan's conjecture for abelian defect groups

TL;DR

This paper introduces the strong Frobenius number, a new invariant for p-blocks, and applies it to prove Donovan's conjecture for certain 2-blocks with abelian defect groups, advancing understanding in modular representation theory.

Contribution

The paper defines the strong Frobenius number and uses it to prove Donovan's conjecture for specific classes of 2-blocks with abelian defect groups.

Findings

Proved Donovan's conjecture for 2-blocks with abelian defect groups of rank ≤ 4.

Confirmed Donovan's conjecture for 2-blocks with abelian defect groups of order ≤ 64.

Introduced the strong Frobenius number as a key tool in block theory.

Abstract

We define a new invariant for a $p$-block, the strong Frobenius number, which we use to address the problem of reducing Donovan's conjecture to normal subgroups of index p. As an application we use the strong Frobenius number to complete the proof of Donovan's conjecture for 2-blocks with abelian defect groups of rank at most 4 and for 2-blocks with abelian defect groups of order at most 64.