Existence of Nonzero Trace-Zero Idempotents in the Group Algebras of Finite Groups

TL;DR

This paper investigates conditions under which trace-zero elements in group algebras of finite groups form Mathieu subspaces and characterizes groups with no nonzero trace-zero idempotents.

Contribution

It provides numerical criteria linking irreducible representation degrees to the existence of trace-zero idempotents in group algebras.

Findings

Numerical conditions for trace-zero subspaces to be Mathieu subspaces

Characterization of groups with no nonzero trace-zero idempotents

Criteria for absence of central trace-zero idempotents

Abstract

Let $G$ be a finite group and $K$ a splitting field of $G$ of characteristic $p>0$. Denote by $KG$ the group algebra of $G$ over $K$ and $Z(KG)$ the center of $KG$. Let $V_G$ be the $K$-subspace of trace-zero elements of $KG$. We give some numerical sufficient and necessary conditions for $V_G$ and $V_G\cap Z(KG)$, respectively, to be Mathieu subspaces of $KG$ in terms of the degrees of irreducible representations of $G$ over $K$. The same numerical conditions also characterize the finite groups $G$ that $KG$ has no nonzero trace-zero idempotents and the finite groups $G$ that $KG$ has no nonzero central trace-zero idempotents, respectively.