Decomposition numbers in the principal block and Sylow normalisers

TL;DR

This paper explores the connection between the modular decomposition numbers of height zero characters in the principal p-block of a finite group G and the local structure of G related to its Sylow p-subgroups.

Contribution

It provides new insights into how the p-local structure influences the decomposition numbers in the principal block of G.

Findings

Established relationships between decomposition numbers and Sylow p-subgroups

Identified conditions under which decomposition numbers can be explicitly determined

Enhanced understanding of the principal block structure in modular representation theory

Abstract

If G is a finite group and p is a prime number, we investigate the relationship between the p-modular decomposition numbers of characters of height zero in the principal p-block of G and the p-local structure of G.