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

Wenhua Zhao; Dan Yan

paper

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

arXiv:2208.06312

Abstract

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

Latex 17 pages