Characterizing varieties of colength ≤4

Abstract

Let A be an associative algebra over a field F of characteristic zero, and let n(A), n=1,2,, be the sequence of cocharacters of A. For every n1, let ln(A) denote the nth colength of A, counting the number of Sn-irreducibles appearing in n(A). In this article, we classify the algebras A such that the sequence of colengths ln(A), n=1,2,, is bounded by four. Moreover we construct a finite number of algebras A1,, Ad, such that ln(A)4 if and only if A1,, Ad var(A).