Blocks with defect group D_{2^n} x C_{2^m}

TL;DR

This paper determines invariants of blocks with defect groups combining dihedral and cyclic groups, proving several key conjectures in modular representation theory for these blocks.

Contribution

It generalizes Brauer's results to blocks with defect groups D_{2^n} x C_{2^m} and proves multiple longstanding conjectures in this context.

Findings

Proves Brauer's k(B)-conjecture for these blocks

Establishes Olsson's and Eaton's conjectures in this setting

Shows the gluing problem has a unique solution

Abstract

We determine the numerical invariants of blocks with defect group D_{2^n}\times C_{2^m}, where D_{2^n} denotes a dihedral group of order 2^n and C_{2^m} denotes a cyclic group of order 2^m. This generalizes Brauer's results for m=0. As a consequence, we prove Brauer's k(B)-conjecture, Olsson's conjecture (and more generally Eaton's conjecture), Brauer's height zero conjecture, the Alperin-McKay conjecture, Alperin's weight conjecture and Robinson's ordinary weight conjecture for these blocks. Moreover, we show that the gluing problem has a unique solution in this case.