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

TL;DR

This paper determines invariants of blocks with a specific defect group structure and proves several major conjectures in block theory for these cases, also establishing the uniqueness of the gluing problem solution.

Contribution

It provides the first complete verification of multiple major conjectures for blocks with defect group D_{2^n} * C_{2^m} and proves the uniqueness of the gluing problem solution in this context.

Findings

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

Verifies Olsson's and other 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} * C_{2^m} = Q_{2^n} * C_{2^m} (central product), where n > 2 and m > 1. 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.