Properties of hook Schur functions with applications to p. i. algebras

We study properties of the coefficients mu(lambda) in infinite series of hook Schur functions Sigma m(lambda)HS(lambda)(x(1), .... x(k); y(1), ... , y(l)) that convergeto rational functions with denominators a product of terms of theform (1 -M), where each M is a monomial in the x(i) and Y-j. As an application, we prove that if A is a p. i. algebra with unit in characteristic 0, then the colength sequence l(n)(A) is asymptotic to a function of the form Cn(t), for some positive real number C and some positive integer t; and the codimension sequence c(n)(A) is asymptotic to a function of the form Cn(t)e(n), for some positive real number C, integer or half-integer t, and positive integer e. (c) 2007 Elsevier Inc. All rights reserved.