Decomposition numbers in the principal block and Sylow normalisers
Gunter Malle, Noelia Rizo
TL;DR
This work connects $p$-modular decomposition numbers $d_{\chi 1_G}$ for height-zero characters in the principal block with the $p$-local structure of finite groups. The authors introduce Property $(*)$ for simple groups and prove Theorem A: for $p>3$, the conditions $d_{\chi 1_G}\neq 0$ for all $\chi\in\mathrm{Irr}_{p'}(B_0(G))$, $d_{\chi 1_G}=1$ for all such $\chi$, and ${\bf N}_G(P)=P\times K$ for all $P\in\mathrm{Syl}_p(G)$ are equivalent, assuming the composition factors satisfy $(*)$. They verify $(*)$ for several families (sporadic, alternating, and many Lie type groups in non-defining characteristic) and provide partial results in defining characteristic, along with a reduction showing the general theorem reduces to verifying $(*)$ on composition factors. The paper also proves the $p$-solvable case of the theorem unconditionally and outlines a comprehensive reduction framework to extend the result to the general case. Overall, the work strengthens the bridge between Brauer character theory and Sylow-like local structural properties, offering concrete criteria and reductions for broad classes of finite groups.
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.