p-Blocks Relative to a Character of a Normal Subgroup

TL;DR

This paper introduces theta-blocks, a new partition of irreducible characters relative to a normal subgroup and prime p, unifying key conjectures in modular representation theory.

Contribution

It defines theta-blocks and theta-defect groups, linking them to major conjectures and providing a new framework to approach longstanding problems.

Findings

Unifies Gluck-Wolf-Navarro-Tiep theorem and Brauer's Height Zero conjecture.

Shows the k(B)-conjecture is equivalent to size conditions on theta-blocks.

Provides a new perspective on the role of normal subgroups in block theory.

Abstract

Let G be a finite group, let N be a normal subgroup of G, and let theta in Irr(N) be a G-invariant character. We fix a prime p, and we introduce a canonical partition of Irr(G|theta) relative to p. We call each member B_theta of this partition a theta-block, and to each theta-block B_theta we naturally associate a conjugacy class of p-subgroups of G/N, which we call the theta-defect groups of B_theta. If N is trivial, then the theta-blocks are the Brauer p-blocks. Using theta-blocks, we can unify the Gluck-Wolf-Navarro-Tiep theorem and Brauer's Height Zero conjecture in a single statement, which, after work of B. Sambale, turns out to be equivalent to the the Height Zero conjecture. We also prove that the k(B)-conjecture is true if and only if every theta-block B_theta has size less than or equal the size of any of its theta-defect groups, hence bringing normal subgroups to the k(B)-conjecture.