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.