On the colength sequence of G-graded algebras

Abstract

Let F be a field of characteristic zero and let A be an Falgebra graded by a finite group G of order k . Given a non-negative integer n and a sum n = n1 1 + + nk k of k non-negative integers, we associate a S ( n )-module to A , where S ( n ) := S n 1 x x Snk, n k , and we denote its S ( n )-character by chi ( n ) ( A ). In this paper, for all sum n = n1 1 + +nk, n k , we make explicit the decomposition of chi ( n ) ( A ) for some important G-graded algebras A and we compute the number lG n(A) G n ( A ) of irreducibles appearing in all such decompositions. Our main goal is to classify G-graded algebras A such that the sequence lGn(A) G n ( A ) is bounded by three.