On the colength sequence of algebras with graded involution

In recent years, many results have been established regarding classifications of varieties whose colength sequences are bounded by a fixed constant. In this work, we explore this theme in the setting of algebras endowed with a graded involution, called $(G,\ast )$-algebras. We give an explicit description of the decomposition of the $⟨n⟩$-cocharacter for some important $(G,\ast )$-algebras $A$, for every $⟨n⟩=(n_1,\dots ,n_{2t})$. For each algebra $A$, the $n$th colength is defined as the number of irreducible components that appear in these decompositions. Our aim is to classify varieties whose $n$th colengths are bounded by a fixed constant.