Decomposition numbers in the principal block and Sylow normalisers

Gunter Malle and Noelia Rizo

Vol. 336 (2025), No. 1-2, 367–378

DOI: 10.2140/pjm.2025.336.367

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 $G$. In particular we prove that, under certain conditions on the nonabelian composition factors of $G$, $d_{\chi 1_G}\ne 0$ for all irreducible characters $\chi$ of degree prime to $p$ in the principal $p$-block of $G$ if, and only if, the normaliser of a Sylow $p$-subgroup of $G$ has a normal $p$-complement.

To the memory of Gary Seitz

Keywords

decomposition numbers, principal blocks, Sylow normalisers

Mathematical Subject Classification

Primary: 20C15, 20C20, 20C33, 20D20

Milestones

Received: 21 May 2024

Revised: 19 July 2024

Accepted: 19 July 2024

Published: 26 May 2025