Colength sequences for Kemer algebras

Abstract

This paper is a sequel to [4]. In that paper we calculated the asymptotic behavior of the colength sequence for k × k matrices, and in this paper we will calculate the asymptotic behavior of the colength sequence for the algebras Mk, . Given a p.i. algebra A, the cocharacter sequence {χn(A)}∞n=0 is a sequence of Sncharacters. If we decompose each character into a sum of irreducible Sn-characters, we would get something of the form χn(A)= ∑λ mλ(A)χ. Let ln(A) be the length of χn(A), so ln(A) = ∑λ∈Par(n) mλ(A). The sequence of numbers {ln(A)} is the colength sequence of A. Similar constructions can also be made for other cocharacter sequences, such as the trace cocharacter if A is an algebra with trace. Kemer classified verbally prime p.i. algebras into three families, see [8]. One is k× k matrices whose colength sequence we investigated in [4]. One is k× k matrices over an infinite dimensional Grassmann algebra. We do not know yet how to deal with this case. And the third is the algebras Mk, . Let E be an infinite dimensional Grassmann algebra and let k + = n. Then Mk, is a subalgebra of the matrix algebra Mn(E). In order to define it, note that the Grassmann algebra has a natural Z/2Z grading, E =E0 +E1 with respect to which it is anticommutative. I.e., degree 0 elements are central and degree 1 elements anticommute with each other. Let i and j be between 1 and n. We define the ordered pair (i, j) to be of degree 0 if either 1 i, j k or k+ 1 i, j n, and of degree 1 otherwise. Then Mk, will be the set of all matrices (aij ) ∈Mn(E) in which the degree of each aij