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=n1+⋯+nk of k non-negative integers, we associate a S〈n〉-module to A, where S〈n〉:=Snjavax.xml.bind.JAXBElement@109e81c8×⋯×Snjavax.xml.bind.JAXBElement@6951e7e6, and we denote its S〈n〉-character by χ〈n〉(A). In this paper, for all sum n=n1+⋯+nk, we make explicit the decomposition of χ〈n〉(A) for some important G-graded algebras A and we compute the number lnG(A) of irreducibles appearing in all such decompositions. Our main goal is to classify G-graded algebras A such that the sequence lnG(A) is bounded by three.