Growth for Algebras Satisfying Polynomial Identities

Abstract

The th codimension of a PI algebra measures how many identities of degree the algebra satisfies. Growth for PI algebras is the rate of growth of as goes to infinity. Since in most cases there is no hope in finding nice closed formula for , we study its asymptotics. We review here such results about , when is an associative PI algebra. We start with the exponential bound on then give few applications. We review some remarkable properties (integer and half integer) of the asymptotics of . The representation theory of the symmetric group is an important tool in this theory.