Brauer's Height Zero Conjecture for metacyclic defect groups

TL;DR

This paper proves Brauer's Height Zero Conjecture for blocks with metacyclic defect groups, verifies related conjectures, and provides new results for specific defect groups, advancing understanding in modular representation theory.

Contribution

It establishes Brauer's Height Zero Conjecture for certain metacyclic defect groups and verifies several related conjectures, including the Alperin-McKay and Alperin's Weight Conjecture, for specific cases.

Findings

Brauer's Height Zero Conjecture holds for metacyclic defect groups.

Verification of the Alperin-McKay Conjecture for p=3.

Proof of the Galois-Alperin-McKay Conjecture for extraspecial defect groups of order p^3.

Abstract

We prove that Brauer's Height Zero Conjecture holds for p-blocks of finite groups with metacyclic defect groups. If the defect group is nonabelian and contains a cyclic maximal subgroup, we obtain the distribution into p-conjugate and p-rational irreducible characters. Then the Alperin-McKay Conjecture follows provided p=3. Along the way we verify a few other conjectures. Finally we consider the extraspecial defect group of order p^3 and exponent p^2 for an odd prime more closely. Here for blocks with inertial index 2 we prove the Galois-Alperin-McKay Conjecture by computing k_0(B). Then for p\le 11 also Alperin's Weight Conjecture follows. This improves some results of [Gao, 2012], [Holloway-Koshitani-Kunugi, 2010] and [Hendren, 2005].