A graph-theoretic approach for comparing dimensions of components in simply-graded algebras

Abstract

Abstract. Any simple group-grading of a finite dimensional complex algebra induces a natural family of digraphs. We prove that |E • E op ∪ E op • E| ≥ |E| for any digraph Γ = (V, E) without parallel edges, and deduce that for any simple group-grading, the dimension of the trivial component is maximal.