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

We study properties of the coefficients mλ in infinite series of hook Schur functions ∑mλHSλ(x1,…,xk;y1,…,yℓ) that converge to rational functions with denominators a product of terms of the form (1−M), where each M is a monomial in the xi and yj. As an application, we prove that if A is a p. i. algebra with unit in characteristic 0, then the colength sequence ln(A) is asymptotic to a function of the form Cnt, for some positive real number C and some positive integer t; and the codimension sequence cn(A) is asymptotic to a function of the form Cnten, for some positive real number C, integer or half-integer t, and positive integer e.