On a conjecture of G. Malle and G. Navarro on nilpotent blocks

TL;DR

This paper verifies Malle and Navarro's conjecture that blocks with uniform height 0 character degrees are exactly the nilpotent blocks, focusing on spin-blocks of the covering group of the alternating group.

Contribution

It confirms the conjecture for spin-blocks of the covering group of the alternating group, addressing an excluded case in previous studies.

Findings

Conjecture holds for spin-blocks of 2.A_n

Provides evidence supporting the conjecture in new cases

Advances understanding of block theory in finite groups

Abstract

In a recent article, G. Malle and G. Navarro conjectured that the $p$-blocks of a finite group all of whose height 0 characters have the same degree are exactly the nilpotent blocks defined by M. Brou\'e and L. Puig. In this paper, we check that this conjecture holds for spin-blocks of the covering group $2.\A_n$ of the alternating group $\A_n$, thereby solving a case excluded from the study of quasi-simple groups by Malle and Navarro.