Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings

TL;DR

This paper reduces Donovan's conjecture for blocks with abelian defect groups over discrete valuation rings to quasisimple groups, establishing new cases and linking the conjecture to bounds on Cartan invariants and Frobenius numbers.

Contribution

It provides a reduction of Donovan's conjecture to quasisimple groups for abelian defect groups over discrete valuation rings, especially for prime two, and connects the conjecture to bounding invariants.

Findings

Donovan's conjecture holds for prime two with abelian defect groups.

Reduction to quasisimple groups for arbitrary primes.

Connection between Donovan's conjecture and bounds on Cartan invariants and Frobenius numbers.

Abstract

We give a reduction to quasisimple groups for Donovan's conjecture for blocks with abelian defect groups defined with respect to a suitable discrete valuation ring $\mathcal{O}$. Consequences are that Donovan's conjecture holds for $\mathcal{O}$-blocks with abelian defect groups for the prime two, and that, using recent work of Farrell and Kessar, for arbitrary primes Donovan's conjecture for $\mathcal{O}$-blocks with abelian defect groups reduces to bounding the Cartan invariants of blocks of quasisimple groups in terms of the defect. A result of independent interest is that in general (i.e. for arbitrary defect groups) Donovan's conjecture for $\mathcal{O}$-blocks is a consequence of conjectures predicting bounds on the $\mathcal{O}$-Frobenius number and on the Cartan invariants, as was proved by Kessar for blocks defined over an algebraically closed field.