Existence of nonzero trace-zero idempotents in the group algebras of finite groups

Abstract

Let [Formula: see text] be a finite group and [Formula: see text] a splitting field of [Formula: see text] of characteristic [Formula: see text]. Denote by [Formula: see text] the group algebra of [Formula: see text] over [Formula: see text] and [Formula: see text] the center of [Formula: see text]. Let [Formula: see text] be the [Formula: see text]-subspace of trace-zero elements of [Formula: see text]. We give some numerical sufficient and necessary conditions for [Formula: see text] and [Formula: see text], respectively, to be Mathieu subspaces of [Formula: see text] in terms of the degrees of irreducible representations of [Formula: see text] over [Formula: see text]. The same numerical conditions also characterize the finite groups [Formula: see text] that [Formula: see text] has no nonzero trace-zero idempotents and the finite groups [Formula: see text] that [Formula: see text] has no nonzero central trace-zero idempotents, respectively.