On integral decomposition of unipotent elements in integral group rings

TL;DR

This paper investigates the conjecture by Jespers and Sun regarding nilpotent elements in integral group rings, characterizing groups with property ND and describing their structure, with almost complete confirmation of the conjecture.

Contribution

The paper proves Jespers and Sun's conjecture for all but one special case, introduces the SN property, and explores related algebraic and representation-theoretic properties.

Findings

Confirmed Jespers and Sun's conjecture up to one exception.

Characterized groups with the SN property.

Connected properties related to the ND property and their equivalence in certain groups.

Abstract

Jespers and Sun conjectured that if a finite group $G$ has the property ND, i.e. for every nilpotent element $n$ in the integral group ring $\mathbb{Z}G$ and every primitive central idempotent $e \in \mathbb{Q}G$ one still has $ne \in \mathbb{Z}G$, then at most one of the simple components of the group algebra $\mathbb{Q} G$ has reduced degree bigger than $1$. With the exception of one very special series of groups we are able to answer their conjecture, showing that it is true - up to exactly one exception. To do so we first describe groups with the so-called SN property which was introduced by Liu and Passman in their investigation of the Multiplicative Jordan Decomposition for integral group rings. We then study further objects connected to the property ND. This concerns on one hand a certain section of the unit group of $\mathbb{Z} G$ which measures how far $G$ is from having ND and about which Jespers and Sun posed two further questions which we answer. On the other hand we introduce two properties which appeared naturally in these investigations: one is purely representation-theoretic, while the other can be regarded as indicating that it might be hard to decide ND. Among others we show these two notions are equivalent for groups with SN.