Principal blocks, irreducible restriction, fields and degrees

TL;DR

This paper investigates the conditions under which characters of almost simple groups belong to the principal block, employing group-theoretical techniques to reduce complex problems to questions about these groups and proposing a Galois analogue of a known conjecture.

Contribution

It introduces a new approach to determine principal block membership for characters of almost simple groups and formulates a Galois analogue of the height-zero-equal-degree conjecture.

Findings

Reduced block problems to questions about almost simple groups.

Formulated a Galois analogue of the height-zero-equal-degree conjecture.

Identified unresolved issues in the 'going up' case of irreducible extensions.

Abstract

Several recent problems in the representation theory of finite groups require determining whether certain characters of almost simple groups belong to the principal block. Since the values of these characters are not yet known, we employ alternative group-theoretical techniques to address the "going down" case. This approach enables us to reduce the block version of well-known results by the third and fourth authors to a question about almost simple groups. Moreover, this suggests a Galois analogue of the height-zero-equal-degree conjecture of Malle and Navarro, which we formulate. However, the "going up" case of irreducible extensions of principal block characters remains unresolved.