Characterization of Certain T-ideals from the view point of representation theory of the Symmetric Groups

Abstract

Let K[X] be a free associative algebra (without identity) over a field K of characteristic 0 with free generators X = (X1, X2, ...), and let Pn be the set of all multilinear elements of degree n in K[X]. Then Pn is a KSn-module, where KSn is the group algebra of the symmetric group Sn. An ideal of K[X] stable under all endomorphisms of K[X] is called a T-ideal. If L is an arbitrary T-ideal of K[X] then Ln := Pn ∩ L is a KSn-module too. An important task in the theory of varieties of algebras is to reveal general regularities in the behavior of the sequence A n for various T-ideals A. In certain cases, given a property P, say, of the sequence, one can find a T-ideal L(P) such that a T-ideal L′ satisfies P if and only if L′ contains L(P). The results of this paper have to be regarded from this point of view.