Clifford theory of characters in induced blocks

TL;DR

This paper introduces a new criterion based on Clifford theory to determine when characters of finite groups extend, providing a character correspondence that generalizes Harris-Kn"orr's theorem and aids in verifying key conjectures in modular representation theory.

Contribution

It develops a novel criterion for character extension in finite groups using Clifford theory, generalizing existing theorems and facilitating the verification of important conjectures.

Findings

Established a new character correspondence under specific group conditions.

Generalized Harris-Kn"orr's theorem in the context of block theory.

Provided tools to check inductive Alperin-McKay and Blockwise Alperin Weight conditions.

Abstract

We present a new criterion to predict if a character of a finite group extends. Let $G$ be a finite group and $p$ a prime. For $N\lhd G$, we consider $p$-blocks $b$ and $b'$ of $N$ and ${\rm N}_N(D)$, respectively, with $(b')^N=b$, where $D$ is a defect group of $b'$. Under the assumption that $G$ coincides with a normal subgroup $G[b]$ of $G$, which was introduced by Dade early 1970's, we give a character correspondence between the sets of all irreducible constituents of $\phi^G$ and those of $(\phi')^{{\rm N}_G(D)}$ where $\phi$ and $\phi'$ are irreducible Brauer characters in $b$ and $b'$, respectively. This implies a sort of generalization of the theorem of Harris-Kn\"orr. An important tool is the existence of certain extensions that also helps in checking the inductive Alperin-McKay and inductive Blockwise Alperin Weight conditions, due to the second author.