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

Abstract

We study properties of the coefficients m λ in infinite series of hook Schur functions ∑ m λ H S λ ( x 1 , … , x k ; y 1 , … , y ℓ ) that converge to rational functions with denominators a product of terms of the form ( 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 C n 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 C n t e n , for some positive real number C, integer or half-integer t, and positive integer e.