On the colength sequence of G-graded algebras

Let F be a field of characteristic zero and let A be an F-algebra graded by a finite group G of order k. Given a non-negative integer n and a sum $n=n_1+\cdots +n_k$ of k non-negative integers, we associate a $S_{〈n〉}$-module to A, where $S_{〈n〉}:=S_{n_1}\times \cdots \times S_{n_k}$, and we denote its $S_{〈n〉}$-character by ${\chi }_{〈n〉}\left(A\right)$. In this paper, for all sum $n=n_1+\cdots +n_k$, we make explicit the decomposition of ${\chi }_{〈n〉}\left(A\right)$ for some important G-graded algebras A and we compute the number $l_n^G\left(A\right)$ of irreducibles appearing in all such decompositions. Our main goal is to classify G-graded algebras A such that the sequence $l_n^G\left(A\right)$ is bounded by three.