Algebras with additional structures and multiplicities bounded by a constant

Let Formula: see text be a finite group and Formula: see text a Formula: see text-graded algebra over a field Formula: see text of characteristic zero. We characterize the varieties of Formula: see text-graded algebras such that the multiplicities Formula: see text appearing in the Formula: see text-cocharacters of Formula: see text are bounded by a constant, in terms of Formula: see text-identities. If Formula: see text is endowed with a graded involution ∗, i.e. if Formula: see text is a Formula: see text-algebra, we characterize the varieties of Formula: see text-algebras whose multiplicities in the sequence of Formula: see text-cocharacters of Formula: see text are bounded by Formula: see text by explicitly providing a list of Formula: see text-polynomial identities satisfied by such varieties.